Skip to content

Reconstruction Operators

High-order flux reconstruction methods (upwind, WENO) for advection terms.

finitevolx.Reconstruction1D

Bases: Module

1-D face-value reconstruction.

Parameters:

Name Type Description Default
grid ArakawaCGrid1D
required
Source code in finitevolx/_src/advection/reconstruction.py 
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
class Reconstruction1D(eqx.Module):
    """1-D face-value reconstruction.

    Parameters
    ----------
    grid : ArakawaCGrid1D
    """

    grid: ArakawaCGrid1D

    def naive_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
    ) -> Float[Array, "Nx"]:
        """Naive (centred) reconstruction at east face.

        fe[i+1/2] = 1/2 * (h[i] + h[i+1]) * u[i+1/2]
        """
        out = interior(0.5 * (h[1:-1] + h[2:]) * u[1:-1], h)
        return out

    def upwind1_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
    ) -> Float[Array, "Nx"]:
        """1st-order upwind reconstruction at east face.

        fe[i+1/2] = h[i]   * u[i+1/2]   if u[i+1/2] >= 0
                  = h[i+1] * u[i+1/2]   otherwise
        """
        # fe[i+1/2] = upwind(h, u) * u[i+1/2]
        h_face = jnp.where(u[1:-1] >= 0.0, h[1:-1], h[2:])
        out = interior(h_face * u[1:-1], h)
        return out

    def upwind2_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
    ) -> Float[Array, "Nx"]:
        """2nd-order upwind reconstruction at east face with boundary fallback.

        Positive flow:  h_face[i+1/2] = 3/2*h[i]   - 1/2*h[i-1]
        Negative flow:  h_face[i+1/2] = 3/2*h[i+1] - 1/2*h[i+2]
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.
        """
        # h_face_pos[i+1/2] = 3/2*h[i] - 1/2*h[i-1]
        h_pos = 1.5 * h[1:-1] - 0.5 * h[:-2]
        # h_face_neg[i+1/2] = 3/2*h[i+1] - 1/2*h[i+2]
        # Use 2nd-order where i+2 is available (all except last interior face)
        h_neg_interior = 1.5 * h[2:-1] - 0.5 * h[3:]
        # Use 1st-order upwind on east boundary face (h_face = h[i+1])
        h_neg_boundary = h[-1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary])
        h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1], h)
        return out

    def upwind3_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
    ) -> Float[Array, "Nx"]:
        """3rd-order upwind reconstruction at east face with boundary fallback.

        Positive flow:  h_face = -1/6*h[i-1] + 5/6*h[i]   + 1/3*h[i+1]
        Negative flow:  h_face =  1/3*h[i]   + 5/6*h[i+1] - 1/6*h[i+2]
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.
        """
        # 3rd-order positive stencil (valid for all interior faces)
        h_pos = (
            -1.0 / 6.0 * h[:-2]  # h[i-1]
            + 5.0 / 6.0 * h[1:-1]  # h[i  ]
            + 1.0 / 3.0 * h[2:]  # h[i+1]
        )
        # 3rd-order negative stencil (valid only where h[i+2] exists)
        # For all interior faces except the last one
        h_neg_interior = (
            1.0 / 3.0 * h[1:-2]  # h[i  ]
            + 5.0 / 6.0 * h[2:-1]  # h[i+1]
            - 1.0 / 6.0 * h[3:]  # h[i+2]
        )
        # 1st-order upwind fallback at east boundary: h_face = h[i+1]
        h_neg_boundary = h[-1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary])
        h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1], h)
        return out

    def weno3_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
    ) -> Float[Array, "Nx"]:
        """3-point WENO east-face flux with boundary fallback.

        Positive flow:  h_face[i+1/2] = WENO3(h[i-1], h[i],   h[i+1])
        Negative flow:  h_face[i+1/2] = WENO3_right(h[i], h[i+1], h[i+2])
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.
        """
        # WENO-3 positive: left-biased stencil, valid for all interior faces
        h_pos = _weno3(h[:-2], h[1:-1], h[2:])
        # WENO-3 negative: right-biased stencil, valid for i+2 < Nx
        h_neg_interior = _weno3_right(h[1:-2], h[2:-1], h[3:])
        # 1st-order upwind fallback at east boundary
        h_neg_boundary = h[-1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary])
        h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1], h)
        return out

    def wenoz3_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
    ) -> Float[Array, "Nx"]:
        """3-point WENO-Z east-face flux with boundary fallback.

        Positive flow:  h_face[i+1/2] = WENOZ3(h[i-1], h[i],   h[i+1])
        Negative flow:  h_face[i+1/2] = WENOZ3_right(h[i], h[i+1], h[i+2])
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.
        """
        # WENO-Z-3 positive: left-biased stencil, valid for all interior faces
        h_pos = _wenoz3(h[:-2], h[1:-1], h[2:])
        # WENO-Z-3 negative: right-biased stencil, valid for i+2 < Nx
        h_neg_interior = _wenoz3_right(h[1:-2], h[2:-1], h[3:])
        # 1st-order upwind fallback at east boundary
        h_neg_boundary = h[-1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary])
        h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1], h)
        return out

    def weno5_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
    ) -> Float[Array, "Nx"]:
        """5-point WENO east-face flux with sign-dependent boundary fallbacks.

        Positive flow:  h_face[i+1/2] = WENO5(h[i-2], h[i-1], h[i], h[i+1], h[i+2])
            for i = 2..Nx-3; WENO3 fallback at i = 1 and i = Nx-2.
        Negative flow:  h_face[i+1/2] = WENO5_right(h[i-1], h[i], h[i+1], h[i+2], h[i+3])
            for i = 1..Nx-4; WENO3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
        """
        # WENO-5 positive: valid for interior faces i=2..Nx-3 (needs h[i-2..i+2])
        h5_pos_interior = _weno5(h[:-4], h[1:-3], h[2:-2], h[3:-1], h[4:])
        # WENO-3 fallback at first interior face (i=1, h[i-2] = h[-1] wraps)
        h3_pos_first = _weno3(h[0:1], h[1:2], h[2:3])
        # WENO-3 fallback at last interior face (i=Nx-2, h[i+2] = h[Nx] out of bounds)
        h3_pos_last = _weno3(h[-3:-2], h[-2:-1], h[-1:])
        h_pos = jnp.concatenate([h3_pos_first, h5_pos_interior, h3_pos_last])
        # WENO-5 right-biased: valid for faces i=1..Nx-4
        h5_neg_interior = _weno5_right(h[:-4], h[1:-3], h[2:-2], h[3:-1], h[4:])
        # WENO-3 right-biased fallback at face i=Nx-3
        h3_neg_penultimate = _weno3_right(h[-3:-2], h[-2:-1], h[-1:])
        # 1st-order upwind fallback at east boundary (i=Nx-2)
        h1_neg_last = h[-1:]
        h_neg = jnp.concatenate([h5_neg_interior, h3_neg_penultimate, h1_neg_last])
        h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1], h)
        return out

    def wenoz5_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
    ) -> Float[Array, "Nx"]:
        """5-point WENO-Z east-face flux with sign-dependent boundary fallbacks.

        Positive flow:  h_face[i+1/2] = WENOZ5(h[i-2], h[i-1], h[i], h[i+1], h[i+2])
            for i = 2..Nx-3; WENO-Z-3 fallback at i = 1 and i = Nx-2.
        Negative flow:  h_face[i+1/2] = WENOZ5_right(h[i-1], h[i], h[i+1], h[i+2], h[i+3])
            for i = 1..Nx-4; WENO-Z-3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
        """
        # WENO-Z-5 positive: valid for interior faces i=2..Nx-3
        h5_pos_interior = _wenoz5(h[:-4], h[1:-3], h[2:-2], h[3:-1], h[4:])
        # WENO-Z-3 fallback at first interior face (i=1)
        h3_pos_first = _wenoz3(h[0:1], h[1:2], h[2:3])
        # WENO-Z-3 fallback at last interior face (i=Nx-2)
        h3_pos_last = _wenoz3(h[-3:-2], h[-2:-1], h[-1:])
        h_pos = jnp.concatenate([h3_pos_first, h5_pos_interior, h3_pos_last])
        # WENO-Z-5 right-biased: valid for faces i=1..Nx-4
        h5_neg_interior = _wenoz5_right(h[:-4], h[1:-3], h[2:-2], h[3:-1], h[4:])
        # WENO-Z-3 right-biased fallback at face i=Nx-3
        h3_neg_penultimate = _wenoz3_right(h[-3:-2], h[-2:-1], h[-1:])
        # 1st-order upwind fallback at east boundary (i=Nx-2)
        h1_neg_last = h[-1:]
        h_neg = jnp.concatenate([h5_neg_interior, h3_neg_penultimate, h1_neg_last])
        h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1], h)
        return out

    def weno7_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
    ) -> Float[Array, "Nx"]:
        """7-point WENO east-face flux with hierarchical boundary fallbacks.

        Positive flow:  h_face[i+1/2] = WENO7(h[i-3..i+3])
            for i = 3..Nx-4; WENO5 fallback at i = 2 and i = Nx-3;
            WENO3 fallback at i = 1 and i = Nx-2.
        Negative flow:  h_face[i+1/2] = WENO7(h[i+3..i-3])
            for i = 3..Nx-4; WENO3 fallback at i = 1; WENO5 fallback
            at i = 2 and i = Nx-3; 1st-order upwind at i = Nx-2.
        """
        return _weno_flux_axis_1d(h, u, order=7)

    def weno9_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
    ) -> Float[Array, "Nx"]:
        """9-point WENO east-face flux with hierarchical boundary fallbacks.

        Positive flow:  h_face[i+1/2] = WENO9(h[i-4..i+4])
            for i = 4..Nx-5; WENO7 fallback at i = 3 and i = Nx-4;
            WENO5 fallback at i = 2 and i = Nx-3; WENO3 fallback at
            i = 1 and i = Nx-2.
        Negative flow:  h_face[i+1/2] = WENO9(h[i+4..i-4])
            for i = 4..Nx-5; WENO7 fallback at i = 3 and i = Nx-4;
            WENO5 fallback at i = 2 and i = Nx-3; WENO3 fallback at
            i = 1; 1st-order upwind at i = Nx-2.
        """
        return _weno_flux_axis_1d(h, u, order=9)

    def tvd_x(
        self,
        h: Float[Array, "Nx"],
        u: Float[Array, "Nx"],
        limiter: str = "minmod",
    ) -> Float[Array, "Nx"]:
        """TVD east-face flux using a flux limiter.

        Blends 1st-order upwind with an anti-diffusive correction limited by
        φ(r) to preserve monotonicity:

        Positive flow:
            r     = (h[i] − h[i−1]) / (h[i+1] − h[i])
            h_L   = h[i]   + ½ φ(r) (h[i+1] − h[i])

        Negative flow:
            r     = (h[i+1] − h[i+2]) / (h[i] − h[i+1])
            h_R   = h[i+1] + ½ φ(r) (h[i]   − h[i+1])
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.

        Parameters
        ----------
        h : Float[Array, "Nx"]
            Scalar at T-points.
        u : Float[Array, "Nx"]
            Velocity at U-points.
        limiter : str
            Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
            or ``'mc'``.

        Returns
        -------
        Float[Array, "Nx"]
            East-face flux with zero ghost entries.
        """
        phi = _get_limiter(limiter)
        # Positive flow: h_face = h[i] + 0.5*phi(r)*(h[i+1]-h[i])
        # r = (h[i]-h[i-1]) / (h[i+1]-h[i])  for i=1..Nx-2
        diff_fwd = h[2:] - h[1:-1]  # h[i+1] - h[i]
        diff_bwd = h[1:-1] - h[:-2]  # h[i] - h[i-1]
        r_pos = diff_bwd / (diff_fwd + _TVD_EPS)
        h_pos = h[1:-1] + 0.5 * phi(r_pos) * diff_fwd
        # Negative flow: h_face = h[i+1] + 0.5*phi(r)*(h[i]-h[i+1])
        # r = (h[i+1]-h[i+2]) / (h[i]-h[i+1])  for i=1..Nx-3
        diff_neg = h[1:-2] - h[2:-1]  # h[i] - h[i+1]
        diff_neg2 = h[2:-1] - h[3:]  # h[i+1] - h[i+2]
        r_neg_int = diff_neg2 / (diff_neg + _TVD_EPS)
        h_neg_interior = h[2:-1] + 0.5 * phi(r_neg_int) * diff_neg
        # 1st-order upwind fallback at east boundary (i=Nx-2)
        h_neg_boundary = h[-1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary])
        h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1], h)
        return out

naive_x(h, u)

Naive (centred) reconstruction at east face.

fe[i+1/2] = 1/2 * (h[i] + h[i+1]) * u[i+1/2]

Source code in finitevolx/_src/advection/reconstruction.py 
293
294
295
296
297
298
299
300
301
302
303
def naive_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
) -> Float[Array, "Nx"]:
    """Naive (centred) reconstruction at east face.

    fe[i+1/2] = 1/2 * (h[i] + h[i+1]) * u[i+1/2]
    """
    out = interior(0.5 * (h[1:-1] + h[2:]) * u[1:-1], h)
    return out

tvd_x(h, u, limiter='minmod')

TVD east-face flux using a flux limiter.

Blends 1st-order upwind with an anti-diffusive correction limited by φ(r) to preserve monotonicity:

Positive flow: r = (h[i] − h[i−1]) / (h[i+1] − h[i]) h_L = h[i] + ½ φ(r) (h[i+1] − h[i])

Negative flow: r = (h[i+1] − h[i+2]) / (h[i] − h[i+1]) h_R = h[i+1] + ½ φ(r) (h[i] − h[i+1]) Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Parameters:

Name Type Description Default
h Float[Array, Nx]

Scalar at T-points.

 required
u Float[Array, Nx]

Velocity at U-points.

 required
limiter str

Flux limiter name: 'minmod', 'van_leer', 'superbee', or 'mc'.

 'minmod'

Returns:

Type Description
Float[Array, Nx]

East-face flux with zero ghost entries.
Source code in finitevolx/_src/advection/reconstruction.py 
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
def tvd_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
    limiter: str = "minmod",
) -> Float[Array, "Nx"]:
    """TVD east-face flux using a flux limiter.

    Blends 1st-order upwind with an anti-diffusive correction limited by
    φ(r) to preserve monotonicity:

    Positive flow:
        r     = (h[i] − h[i−1]) / (h[i+1] − h[i])
        h_L   = h[i]   + ½ φ(r) (h[i+1] − h[i])

    Negative flow:
        r     = (h[i+1] − h[i+2]) / (h[i] − h[i+1])
        h_R   = h[i+1] + ½ φ(r) (h[i]   − h[i+1])
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.

    Parameters
    ----------
    h : Float[Array, "Nx"]
        Scalar at T-points.
    u : Float[Array, "Nx"]
        Velocity at U-points.
    limiter : str
        Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
        or ``'mc'``.

    Returns
    -------
    Float[Array, "Nx"]
        East-face flux with zero ghost entries.
    """
    phi = _get_limiter(limiter)
    # Positive flow: h_face = h[i] + 0.5*phi(r)*(h[i+1]-h[i])
    # r = (h[i]-h[i-1]) / (h[i+1]-h[i])  for i=1..Nx-2
    diff_fwd = h[2:] - h[1:-1]  # h[i+1] - h[i]
    diff_bwd = h[1:-1] - h[:-2]  # h[i] - h[i-1]
    r_pos = diff_bwd / (diff_fwd + _TVD_EPS)
    h_pos = h[1:-1] + 0.5 * phi(r_pos) * diff_fwd
    # Negative flow: h_face = h[i+1] + 0.5*phi(r)*(h[i]-h[i+1])
    # r = (h[i+1]-h[i+2]) / (h[i]-h[i+1])  for i=1..Nx-3
    diff_neg = h[1:-2] - h[2:-1]  # h[i] - h[i+1]
    diff_neg2 = h[2:-1] - h[3:]  # h[i+1] - h[i+2]
    r_neg_int = diff_neg2 / (diff_neg + _TVD_EPS)
    h_neg_interior = h[2:-1] + 0.5 * phi(r_neg_int) * diff_neg
    # 1st-order upwind fallback at east boundary (i=Nx-2)
    h_neg_boundary = h[-1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary])
    h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1], h)
    return out

upwind1_x(h, u)

1st-order upwind reconstruction at east face.

fe[i+1/2] = h[i] * u[i+1/2] if u[i+1/2] >= 0 = h[i+1] * u[i+1/2] otherwise

Source code in finitevolx/_src/advection/reconstruction.py 
305
306
307
308
309
310
311
312
313
314
315
316
317
318
def upwind1_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
) -> Float[Array, "Nx"]:
    """1st-order upwind reconstruction at east face.

    fe[i+1/2] = h[i]   * u[i+1/2]   if u[i+1/2] >= 0
              = h[i+1] * u[i+1/2]   otherwise
    """
    # fe[i+1/2] = upwind(h, u) * u[i+1/2]
    h_face = jnp.where(u[1:-1] >= 0.0, h[1:-1], h[2:])
    out = interior(h_face * u[1:-1], h)
    return out

upwind2_x(h, u)

2nd-order upwind reconstruction at east face with boundary fallback.

Positive flow: h_face[i+1/2] = 3/2h[i] - 1/2h[i-1] Negative flow: h_face[i+1/2] = 3/2h[i+1] - 1/2h[i+2] Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
def upwind2_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
) -> Float[Array, "Nx"]:
    """2nd-order upwind reconstruction at east face with boundary fallback.

    Positive flow:  h_face[i+1/2] = 3/2*h[i]   - 1/2*h[i-1]
    Negative flow:  h_face[i+1/2] = 3/2*h[i+1] - 1/2*h[i+2]
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.
    """
    # h_face_pos[i+1/2] = 3/2*h[i] - 1/2*h[i-1]
    h_pos = 1.5 * h[1:-1] - 0.5 * h[:-2]
    # h_face_neg[i+1/2] = 3/2*h[i+1] - 1/2*h[i+2]
    # Use 2nd-order where i+2 is available (all except last interior face)
    h_neg_interior = 1.5 * h[2:-1] - 0.5 * h[3:]
    # Use 1st-order upwind on east boundary face (h_face = h[i+1])
    h_neg_boundary = h[-1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary])
    h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1], h)
    return out

upwind3_x(h, u)

3rd-order upwind reconstruction at east face with boundary fallback.

Positive flow: h_face = -1/6h[i-1] + 5/6h[i] + 1/3h[i+1] Negative flow: h_face = 1/3h[i] + 5/6h[i+1] - 1/6h[i+2] Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
def upwind3_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
) -> Float[Array, "Nx"]:
    """3rd-order upwind reconstruction at east face with boundary fallback.

    Positive flow:  h_face = -1/6*h[i-1] + 5/6*h[i]   + 1/3*h[i+1]
    Negative flow:  h_face =  1/3*h[i]   + 5/6*h[i+1] - 1/6*h[i+2]
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.
    """
    # 3rd-order positive stencil (valid for all interior faces)
    h_pos = (
        -1.0 / 6.0 * h[:-2]  # h[i-1]
        + 5.0 / 6.0 * h[1:-1]  # h[i  ]
        + 1.0 / 3.0 * h[2:]  # h[i+1]
    )
    # 3rd-order negative stencil (valid only where h[i+2] exists)
    # For all interior faces except the last one
    h_neg_interior = (
        1.0 / 3.0 * h[1:-2]  # h[i  ]
        + 5.0 / 6.0 * h[2:-1]  # h[i+1]
        - 1.0 / 6.0 * h[3:]  # h[i+2]
    )
    # 1st-order upwind fallback at east boundary: h_face = h[i+1]
    h_neg_boundary = h[-1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary])
    h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1], h)
    return out

weno3_x(h, u)

3-point WENO east-face flux with boundary fallback.

Positive flow: h_face[i+1/2] = WENO3(h[i-1], h[i], h[i+1]) Negative flow: h_face[i+1/2] = WENO3_right(h[i], h[i+1], h[i+2]) Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
def weno3_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
) -> Float[Array, "Nx"]:
    """3-point WENO east-face flux with boundary fallback.

    Positive flow:  h_face[i+1/2] = WENO3(h[i-1], h[i],   h[i+1])
    Negative flow:  h_face[i+1/2] = WENO3_right(h[i], h[i+1], h[i+2])
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.
    """
    # WENO-3 positive: left-biased stencil, valid for all interior faces
    h_pos = _weno3(h[:-2], h[1:-1], h[2:])
    # WENO-3 negative: right-biased stencil, valid for i+2 < Nx
    h_neg_interior = _weno3_right(h[1:-2], h[2:-1], h[3:])
    # 1st-order upwind fallback at east boundary
    h_neg_boundary = h[-1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary])
    h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1], h)
    return out

weno5_x(h, u)

5-point WENO east-face flux with sign-dependent boundary fallbacks.

Positive flow: h_face[i+1/2] = WENO5(h[i-2], h[i-1], h[i], h[i+1], h[i+2]) for i = 2..Nx-3; WENO3 fallback at i = 1 and i = Nx-2. Negative flow: h_face[i+1/2] = WENO5_right(h[i-1], h[i], h[i+1], h[i+2], h[i+3]) for i = 1..Nx-4; WENO3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.

Source code in finitevolx/_src/advection/reconstruction.py 
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
def weno5_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
) -> Float[Array, "Nx"]:
    """5-point WENO east-face flux with sign-dependent boundary fallbacks.

    Positive flow:  h_face[i+1/2] = WENO5(h[i-2], h[i-1], h[i], h[i+1], h[i+2])
        for i = 2..Nx-3; WENO3 fallback at i = 1 and i = Nx-2.
    Negative flow:  h_face[i+1/2] = WENO5_right(h[i-1], h[i], h[i+1], h[i+2], h[i+3])
        for i = 1..Nx-4; WENO3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
    """
    # WENO-5 positive: valid for interior faces i=2..Nx-3 (needs h[i-2..i+2])
    h5_pos_interior = _weno5(h[:-4], h[1:-3], h[2:-2], h[3:-1], h[4:])
    # WENO-3 fallback at first interior face (i=1, h[i-2] = h[-1] wraps)
    h3_pos_first = _weno3(h[0:1], h[1:2], h[2:3])
    # WENO-3 fallback at last interior face (i=Nx-2, h[i+2] = h[Nx] out of bounds)
    h3_pos_last = _weno3(h[-3:-2], h[-2:-1], h[-1:])
    h_pos = jnp.concatenate([h3_pos_first, h5_pos_interior, h3_pos_last])
    # WENO-5 right-biased: valid for faces i=1..Nx-4
    h5_neg_interior = _weno5_right(h[:-4], h[1:-3], h[2:-2], h[3:-1], h[4:])
    # WENO-3 right-biased fallback at face i=Nx-3
    h3_neg_penultimate = _weno3_right(h[-3:-2], h[-2:-1], h[-1:])
    # 1st-order upwind fallback at east boundary (i=Nx-2)
    h1_neg_last = h[-1:]
    h_neg = jnp.concatenate([h5_neg_interior, h3_neg_penultimate, h1_neg_last])
    h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1], h)
    return out

weno7_x(h, u)

7-point WENO east-face flux with hierarchical boundary fallbacks.

Positive flow: h_face[i+1/2] = WENO7(h[i-3..i+3]) for i = 3..Nx-4; WENO5 fallback at i = 2 and i = Nx-3; WENO3 fallback at i = 1 and i = Nx-2. Negative flow: h_face[i+1/2] = WENO7(h[i+3..i-3]) for i = 3..Nx-4; WENO3 fallback at i = 1; WENO5 fallback at i = 2 and i = Nx-3; 1st-order upwind at i = Nx-2.

Source code in finitevolx/_src/advection/reconstruction.py 
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
def weno7_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
) -> Float[Array, "Nx"]:
    """7-point WENO east-face flux with hierarchical boundary fallbacks.

    Positive flow:  h_face[i+1/2] = WENO7(h[i-3..i+3])
        for i = 3..Nx-4; WENO5 fallback at i = 2 and i = Nx-3;
        WENO3 fallback at i = 1 and i = Nx-2.
    Negative flow:  h_face[i+1/2] = WENO7(h[i+3..i-3])
        for i = 3..Nx-4; WENO3 fallback at i = 1; WENO5 fallback
        at i = 2 and i = Nx-3; 1st-order upwind at i = Nx-2.
    """
    return _weno_flux_axis_1d(h, u, order=7)

weno9_x(h, u)

9-point WENO east-face flux with hierarchical boundary fallbacks.

Positive flow: h_face[i+1/2] = WENO9(h[i-4..i+4]) for i = 4..Nx-5; WENO7 fallback at i = 3 and i = Nx-4; WENO5 fallback at i = 2 and i = Nx-3; WENO3 fallback at i = 1 and i = Nx-2. Negative flow: h_face[i+1/2] = WENO9(h[i+4..i-4]) for i = 4..Nx-5; WENO7 fallback at i = 3 and i = Nx-4; WENO5 fallback at i = 2 and i = Nx-3; WENO3 fallback at i = 1; 1st-order upwind at i = Nx-2.

Source code in finitevolx/_src/advection/reconstruction.py 
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
def weno9_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
) -> Float[Array, "Nx"]:
    """9-point WENO east-face flux with hierarchical boundary fallbacks.

    Positive flow:  h_face[i+1/2] = WENO9(h[i-4..i+4])
        for i = 4..Nx-5; WENO7 fallback at i = 3 and i = Nx-4;
        WENO5 fallback at i = 2 and i = Nx-3; WENO3 fallback at
        i = 1 and i = Nx-2.
    Negative flow:  h_face[i+1/2] = WENO9(h[i+4..i-4])
        for i = 4..Nx-5; WENO7 fallback at i = 3 and i = Nx-4;
        WENO5 fallback at i = 2 and i = Nx-3; WENO3 fallback at
        i = 1; 1st-order upwind at i = Nx-2.
    """
    return _weno_flux_axis_1d(h, u, order=9)

wenoz3_x(h, u)

3-point WENO-Z east-face flux with boundary fallback.

Positive flow: h_face[i+1/2] = WENOZ3(h[i-1], h[i], h[i+1]) Negative flow: h_face[i+1/2] = WENOZ3_right(h[i], h[i+1], h[i+2]) Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
def wenoz3_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
) -> Float[Array, "Nx"]:
    """3-point WENO-Z east-face flux with boundary fallback.

    Positive flow:  h_face[i+1/2] = WENOZ3(h[i-1], h[i],   h[i+1])
    Negative flow:  h_face[i+1/2] = WENOZ3_right(h[i], h[i+1], h[i+2])
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.
    """
    # WENO-Z-3 positive: left-biased stencil, valid for all interior faces
    h_pos = _wenoz3(h[:-2], h[1:-1], h[2:])
    # WENO-Z-3 negative: right-biased stencil, valid for i+2 < Nx
    h_neg_interior = _wenoz3_right(h[1:-2], h[2:-1], h[3:])
    # 1st-order upwind fallback at east boundary
    h_neg_boundary = h[-1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary])
    h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1], h)
    return out

wenoz5_x(h, u)

5-point WENO-Z east-face flux with sign-dependent boundary fallbacks.

Positive flow: h_face[i+1/2] = WENOZ5(h[i-2], h[i-1], h[i], h[i+1], h[i+2]) for i = 2..Nx-3; WENO-Z-3 fallback at i = 1 and i = Nx-2. Negative flow: h_face[i+1/2] = WENOZ5_right(h[i-1], h[i], h[i+1], h[i+2], h[i+3]) for i = 1..Nx-4; WENO-Z-3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.

Source code in finitevolx/_src/advection/reconstruction.py 
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
def wenoz5_x(
    self,
    h: Float[Array, "Nx"],
    u: Float[Array, "Nx"],
) -> Float[Array, "Nx"]:
    """5-point WENO-Z east-face flux with sign-dependent boundary fallbacks.

    Positive flow:  h_face[i+1/2] = WENOZ5(h[i-2], h[i-1], h[i], h[i+1], h[i+2])
        for i = 2..Nx-3; WENO-Z-3 fallback at i = 1 and i = Nx-2.
    Negative flow:  h_face[i+1/2] = WENOZ5_right(h[i-1], h[i], h[i+1], h[i+2], h[i+3])
        for i = 1..Nx-4; WENO-Z-3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
    """
    # WENO-Z-5 positive: valid for interior faces i=2..Nx-3
    h5_pos_interior = _wenoz5(h[:-4], h[1:-3], h[2:-2], h[3:-1], h[4:])
    # WENO-Z-3 fallback at first interior face (i=1)
    h3_pos_first = _wenoz3(h[0:1], h[1:2], h[2:3])
    # WENO-Z-3 fallback at last interior face (i=Nx-2)
    h3_pos_last = _wenoz3(h[-3:-2], h[-2:-1], h[-1:])
    h_pos = jnp.concatenate([h3_pos_first, h5_pos_interior, h3_pos_last])
    # WENO-Z-5 right-biased: valid for faces i=1..Nx-4
    h5_neg_interior = _wenoz5_right(h[:-4], h[1:-3], h[2:-2], h[3:-1], h[4:])
    # WENO-Z-3 right-biased fallback at face i=Nx-3
    h3_neg_penultimate = _wenoz3_right(h[-3:-2], h[-2:-1], h[-1:])
    # 1st-order upwind fallback at east boundary (i=Nx-2)
    h1_neg_last = h[-1:]
    h_neg = jnp.concatenate([h5_neg_interior, h3_neg_penultimate, h1_neg_last])
    h_face = jnp.where(u[1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1], h)
    return out

finitevolx.Reconstruction2D

Bases: Module

2-D face-value reconstruction for advective fluxes.

Parameters:

Name Type Description Default
grid ArakawaCGrid2D
required
Source code in finitevolx/_src/advection/reconstruction.py 
 568
 569
 570
 571
 572
 573
 574
 575
 576
 577
 578
 579
 580
 581
 582
 583
 584
 585
 586
 587
 588
 589
 590
 591
 592
 593
 594
 595
 596
 597
 598
 599
 600
 601
 602
 603
 604
 605
 606
 607
 608
 609
 610
 611
 612
 613
 614
 615
 616
 617
 618
 619
 620
 621
 622
 623
 624
 625
 626
 627
 628
 629
 630
 631
 632
 633
 634
 635
 636
 637
 638
 639
 640
 641
 642
 643
 644
 645
 646
 647
 648
 649
 650
 651
 652
 653
 654
 655
 656
 657
 658
 659
 660
 661
 662
 663
 664
 665
 666
 667
 668
 669
 670
 671
 672
 673
 674
 675
 676
 677
 678
 679
 680
 681
 682
 683
 684
 685
 686
 687
 688
 689
 690
 691
 692
 693
 694
 695
 696
 697
 698
 699
 700
 701
 702
 703
 704
 705
 706
 707
 708
 709
 710
 711
 712
 713
 714
 715
 716
 717
 718
 719
 720
 721
 722
 723
 724
 725
 726
 727
 728
 729
 730
 731
 732
 733
 734
 735
 736
 737
 738
 739
 740
 741
 742
 743
 744
 745
 746
 747
 748
 749
 750
 751
 752
 753
 754
 755
 756
 757
 758
 759
 760
 761
 762
 763
 764
 765
 766
 767
 768
 769
 770
 771
 772
 773
 774
 775
 776
 777
 778
 779
 780
 781
 782
 783
 784
 785
 786
 787
 788
 789
 790
 791
 792
 793
 794
 795
 796
 797
 798
 799
 800
 801
 802
 803
 804
 805
 806
 807
 808
 809
 810
 811
 812
 813
 814
 815
 816
 817
 818
 819
 820
 821
 822
 823
 824
 825
 826
 827
 828
 829
 830
 831
 832
 833
 834
 835
 836
 837
 838
 839
 840
 841
 842
 843
 844
 845
 846
 847
 848
 849
 850
 851
 852
 853
 854
 855
 856
 857
 858
 859
 860
 861
 862
 863
 864
 865
 866
 867
 868
 869
 870
 871
 872
 873
 874
 875
 876
 877
 878
 879
 880
 881
 882
 883
 884
 885
 886
 887
 888
 889
 890
 891
 892
 893
 894
 895
 896
 897
 898
 899
 900
 901
 902
 903
 904
 905
 906
 907
 908
 909
 910
 911
 912
 913
 914
 915
 916
 917
 918
 919
 920
 921
 922
 923
 924
 925
 926
 927
 928
 929
 930
 931
 932
 933
 934
 935
 936
 937
 938
 939
 940
 941
 942
 943
 944
 945
 946
 947
 948
 949
 950
 951
 952
 953
 954
 955
 956
 957
 958
 959
 960
 961
 962
 963
 964
 965
 966
 967
 968
 969
 970
 971
 972
 973
 974
 975
 976
 977
 978
 979
 980
 981
 982
 983
 984
 985
 986
 987
 988
 989
 990
 991
 992
 993
 994
 995
 996
 997
 998
 999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
class Reconstruction2D(eqx.Module):
    """2-D face-value reconstruction for advective fluxes.

    Parameters
    ----------
    grid : ArakawaCGrid2D
    """

    grid: ArakawaCGrid2D

    def naive_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """Naive (centred) east-face flux.

        fe[j, i+1/2] = 1/2 * (h[j, i] + h[j, i+1]) * u[j, i+1/2]
        """
        out = interior(0.5 * (h[1:-1, 1:-1] + h[1:-1, 2:]) * u[1:-1, 1:-1], h)
        return out

    def naive_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """Naive (centred) north-face flux.

        fn[j+1/2, i] = 1/2 * (h[j, i] + h[j+1, i]) * v[j+1/2, i]
        """
        out = interior(0.5 * (h[1:-1, 1:-1] + h[2:, 1:-1]) * v[1:-1, 1:-1], h)
        return out

    def upwind1_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """1st-order upwind east-face flux.

        fe[j, i+1/2] = h[j, i]   * u  if u >= 0
                     = h[j, i+1] * u  otherwise
        """
        # fe[j, i+1/2] = upwind1(h, u) * u[j, i+1/2]
        h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h[1:-1, 1:-1], h[1:-1, 2:])
        out = interior(h_face * u[1:-1, 1:-1], h)
        return out

    def upwind1_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """1st-order upwind north-face flux.

        fn[j+1/2, i] = h[j, i]   * v  if v >= 0
                     = h[j+1, i] * v  otherwise
        """
        # fn[j+1/2, i] = upwind1(h, v) * v[j+1/2, i]
        h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h[1:-1, 1:-1], h[2:, 1:-1])
        out = interior(h_face * v[1:-1, 1:-1], h)
        return out

    def upwind2_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """2nd-order upwind east-face flux with boundary fallback.

        Positive: h_face = 3/2*h[j,i] - 1/2*h[j,i-1]
        Negative: h_face = 3/2*h[j,i+1] - 1/2*h[j,i+2]
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.
        """
        # 2nd-order positive stencil (uses h[j,i-1])
        h_pos = 1.5 * h[1:-1, 1:-1] - 0.5 * h[1:-1, :-2]
        # 2nd-order negative stencil (uses h[j,i+2])
        # Use 2nd-order where i+2 is available (all except last interior column)
        h_neg_interior = 1.5 * h[1:-1, 2:-1] - 0.5 * h[1:-1, 3:]
        # Use 1st-order upwind on east boundary column (h_face = h[j,i+1])
        h_neg_boundary = h[1:-1, -1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
        h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1], h)
        return out

    def upwind2_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """2nd-order upwind north-face flux with boundary fallback.

        Positive: h_face = 3/2*h[j,i] - 1/2*h[j-1,i]
        Negative: h_face = 3/2*h[j+1,i] - 1/2*h[j+2,i]
        Falls back to 1st-order upwind on north boundary where j+2 unavailable.
        """
        # 2nd-order positive stencil (uses h[j-1,i])
        h_pos = 1.5 * h[1:-1, 1:-1] - 0.5 * h[:-2, 1:-1]
        # 2nd-order negative stencil (uses h[j+2,i])
        # Use 2nd-order where j+2 is available (all except last interior row)
        h_neg_interior = 1.5 * h[2:-1, 1:-1] - 0.5 * h[3:, 1:-1]
        # Use 1st-order upwind on north boundary row (h_face = h[j+1,i])
        h_neg_boundary = h[-1:, 1:-1]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=0)
        h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1], h)
        return out

    def upwind3_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """3rd-order upwind east-face flux with boundary fallback.

        Positive: h_face = -1/6*h[j,i-1] + 5/6*h[j,i]   + 1/3*h[j,i+1]
        Negative: h_face =  1/3*h[j,i]   + 5/6*h[j,i+1] - 1/6*h[j,i+2]
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.
        """
        # 3rd-order positive stencil (valid for all interior faces)
        h_pos = (
            -1.0 / 6.0 * h[1:-1, :-2]  # h[j, i-1]
            + 5.0 / 6.0 * h[1:-1, 1:-1]  # h[j, i  ]
            + 1.0 / 3.0 * h[1:-1, 2:]  # h[j, i+1]
        )
        # 3rd-order negative stencil (valid only where h[j,i+2] exists)
        # For all interior columns except the last one
        h_neg_interior = (
            1.0 / 3.0 * h[1:-1, 1:-2]  # h[j, i  ]
            + 5.0 / 6.0 * h[1:-1, 2:-1]  # h[j, i+1]
            - 1.0 / 6.0 * h[1:-1, 3:]  # h[j, i+2]
        )
        # 1st-order upwind fallback at east boundary: h_face = h[j,i+1]
        h_neg_boundary = h[1:-1, -1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
        h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1], h)
        return out

    def upwind3_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """3rd-order upwind north-face flux with boundary fallback.

        Positive: h_face = -1/6*h[j-1,i] + 5/6*h[j,i]   + 1/3*h[j+1,i]
        Negative: h_face =  1/3*h[j,i]   + 5/6*h[j+1,i] - 1/6*h[j+2,i]
        Falls back to 1st-order upwind on north boundary where j+2 unavailable.
        """
        # 3rd-order positive stencil (valid for all interior faces)
        h_pos = (
            -1.0 / 6.0 * h[:-2, 1:-1]  # h[j-1, i]
            + 5.0 / 6.0 * h[1:-1, 1:-1]  # h[j,   i]
            + 1.0 / 3.0 * h[2:, 1:-1]  # h[j+1, i]
        )
        # 3rd-order negative stencil (valid only where h[j+2,i] exists)
        # For all interior rows except the last one
        h_neg_interior = (
            1.0 / 3.0 * h[1:-2, 1:-1]  # h[j,   i]
            + 5.0 / 6.0 * h[2:-1, 1:-1]  # h[j+1, i]
            - 1.0 / 6.0 * h[3:, 1:-1]  # h[j+2, i]
        )
        # 1st-order upwind fallback at north boundary: h_face = h[j+1,i]
        h_neg_boundary = h[-1:, 1:-1]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=0)
        h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1], h)
        return out

    def weno3_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """3-point WENO east-face flux with boundary fallback.

        Positive flow:  fe[j,i+1/2] = WENO3(h[j,i-1], h[j,i],   h[j,i+1]) * u
        Negative flow:  fe[j,i+1/2] = WENO3_right(h[j,i], h[j,i+1], h[j,i+2]) * u
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.
        """
        # WENO-3 positive: left-biased, valid for all interior faces
        h_pos = _weno3(h[1:-1, :-2], h[1:-1, 1:-1], h[1:-1, 2:])
        # WENO-3 negative: right-biased, valid for i+2 < Nx
        h_neg_interior = _weno3_right(h[1:-1, 1:-2], h[1:-1, 2:-1], h[1:-1, 3:])
        # 1st-order upwind fallback at east boundary column
        h_neg_boundary = h[1:-1, -1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
        h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1], h)
        return out

    def weno3_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """3-point WENO north-face flux with boundary fallback.

        Positive flow:  fn[j+1/2,i] = WENO3(h[j-1,i], h[j,i],   h[j+1,i]) * v
        Negative flow:  fn[j+1/2,i] = WENO3_right(h[j,i], h[j+1,i], h[j+2,i]) * v
        Falls back to 1st-order upwind on north boundary where j+2 unavailable.
        """
        # WENO-3 positive: left-biased, valid for all interior faces
        h_pos = _weno3(h[:-2, 1:-1], h[1:-1, 1:-1], h[2:, 1:-1])
        # WENO-3 negative: right-biased, valid for j+2 < Ny
        h_neg_interior = _weno3_right(h[1:-2, 1:-1], h[2:-1, 1:-1], h[3:, 1:-1])
        # 1st-order upwind fallback at north boundary row
        h_neg_boundary = h[-1:, 1:-1]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=0)
        h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1], h)
        return out

    def wenoz3_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """3-point WENO-Z east-face flux with boundary fallback.

        Positive flow:  fe[j,i+1/2] = WENOZ3(h[j,i-1], h[j,i],   h[j,i+1]) * u
        Negative flow:  fe[j,i+1/2] = WENOZ3_right(h[j,i], h[j,i+1], h[j,i+2]) * u
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.
        """
        # WENO-Z-3 positive: left-biased, valid for all interior faces
        h_pos = _wenoz3(h[1:-1, :-2], h[1:-1, 1:-1], h[1:-1, 2:])
        # WENO-Z-3 negative: right-biased, valid for i+2 < Nx
        h_neg_interior = _wenoz3_right(h[1:-1, 1:-2], h[1:-1, 2:-1], h[1:-1, 3:])
        # 1st-order upwind fallback at east boundary column
        h_neg_boundary = h[1:-1, -1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
        h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1], h)
        return out

    def wenoz3_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """3-point WENO-Z north-face flux with boundary fallback.

        Positive flow:  fn[j+1/2,i] = WENOZ3(h[j-1,i], h[j,i],   h[j+1,i]) * v
        Negative flow:  fn[j+1/2,i] = WENOZ3_right(h[j,i], h[j+1,i], h[j+2,i]) * v
        Falls back to 1st-order upwind on north boundary where j+2 unavailable.
        """
        # WENO-Z-3 positive: left-biased, valid for all interior faces
        h_pos = _wenoz3(h[:-2, 1:-1], h[1:-1, 1:-1], h[2:, 1:-1])
        # WENO-Z-3 negative: right-biased, valid for j+2 < Ny
        h_neg_interior = _wenoz3_right(h[1:-2, 1:-1], h[2:-1, 1:-1], h[3:, 1:-1])
        # 1st-order upwind fallback at north boundary row
        h_neg_boundary = h[-1:, 1:-1]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=0)
        h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1], h)
        return out

    def weno5_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """5-point WENO east-face flux with sign-dependent boundary fallbacks.

        Positive flow:  fe[j,i+1/2] = WENO5(h[j,i-2..i+2]) * u
            for i = 2..Nx-3; WENO3 fallback at i = 1 and i = Nx-2.
        Negative flow:  fe[j,i+1/2] = WENO5_right(h[j,i-1], h[j,i], h[j,i+1], h[j,i+2], h[j,i+3]) * u
            for i = 1..Nx-4; WENO3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
        """
        # WENO-5 positive: valid for i=2..Nx-3
        h5_pos_int = _weno5(
            h[1:-1, :-4], h[1:-1, 1:-3], h[1:-1, 2:-2], h[1:-1, 3:-1], h[1:-1, 4:]
        )
        # WENO-3 fallback at first interior column (i=1)
        h3_pos_first = _weno3(h[1:-1, 0:1], h[1:-1, 1:2], h[1:-1, 2:3])
        # WENO-3 fallback at last interior column (i=Nx-2)
        h3_pos_last = _weno3(h[1:-1, -3:-2], h[1:-1, -2:-1], h[1:-1, -1:])
        h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=1)
        # WENO-5 right-biased: valid for i=1..Nx-4
        h5_neg_int = _weno5_right(
            h[1:-1, :-4], h[1:-1, 1:-3], h[1:-1, 2:-2], h[1:-1, 3:-1], h[1:-1, 4:]
        )
        # WENO-3 right-biased fallback at column i=Nx-3
        h3_neg_penultimate = _weno3_right(h[1:-1, -3:-2], h[1:-1, -2:-1], h[1:-1, -1:])
        # 1st-order upwind fallback at east boundary column (i=Nx-2)
        h1_neg_last = h[1:-1, -1:]
        h_neg = jnp.concatenate([h5_neg_int, h3_neg_penultimate, h1_neg_last], axis=1)
        h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1], h)
        return out

    def weno5_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """5-point WENO north-face flux with sign-dependent boundary fallbacks.

        Positive flow:  fn[j+1/2,i] = WENO5(h[j-2..j+2,i]) * v
            for j = 2..Ny-3; WENO3 fallback at j = 1 and j = Ny-2.
        Negative flow:  fn[j+1/2,i] = WENO5_right(h[j-1,i], h[j,i], h[j+1,i], h[j+2,i], h[j+3,i]) * v
            for j = 1..Ny-4; WENO3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.
        """
        # WENO-5 positive: valid for j=2..Ny-3
        h5_pos_int = _weno5(
            h[:-4, 1:-1], h[1:-3, 1:-1], h[2:-2, 1:-1], h[3:-1, 1:-1], h[4:, 1:-1]
        )
        # WENO-3 fallback at first interior row (j=1)
        h3_pos_first = _weno3(h[0:1, 1:-1], h[1:2, 1:-1], h[2:3, 1:-1])
        # WENO-3 fallback at last interior row (j=Ny-2)
        h3_pos_last = _weno3(h[-3:-2, 1:-1], h[-2:-1, 1:-1], h[-1:, 1:-1])
        h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=0)
        # WENO-5 right-biased: valid for j=1..Ny-4
        h5_neg_int = _weno5_right(
            h[:-4, 1:-1], h[1:-3, 1:-1], h[2:-2, 1:-1], h[3:-1, 1:-1], h[4:, 1:-1]
        )
        # WENO-3 right-biased fallback at row j=Ny-3
        h3_neg_penultimate = _weno3_right(h[-3:-2, 1:-1], h[-2:-1, 1:-1], h[-1:, 1:-1])
        # 1st-order upwind fallback at north boundary row (j=Ny-2)
        h1_neg_last = h[-1:, 1:-1]
        h_neg = jnp.concatenate([h5_neg_int, h3_neg_penultimate, h1_neg_last], axis=0)
        h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1], h)
        return out

    def wenoz5_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """5-point WENO-Z east-face flux with sign-dependent boundary fallbacks.

        Positive flow:  fe[j,i+1/2] = WENOZ5(h[j,i-2..i+2]) * u
            for i = 2..Nx-3; WENO-Z-3 fallback at i = 1 and i = Nx-2.
        Negative flow:  fe[j,i+1/2] = WENOZ5_right(h[j,i-1], h[j,i], h[j,i+1], h[j,i+2], h[j,i+3]) * u
            for i = 1..Nx-4; WENO-Z-3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
        """
        # WENO-Z-5 positive: valid for i=2..Nx-3
        h5_pos_int = _wenoz5(
            h[1:-1, :-4], h[1:-1, 1:-3], h[1:-1, 2:-2], h[1:-1, 3:-1], h[1:-1, 4:]
        )
        # WENO-Z-3 fallback at first interior column (i=1)
        h3_pos_first = _wenoz3(h[1:-1, 0:1], h[1:-1, 1:2], h[1:-1, 2:3])
        # WENO-Z-3 fallback at last interior column (i=Nx-2)
        h3_pos_last = _wenoz3(h[1:-1, -3:-2], h[1:-1, -2:-1], h[1:-1, -1:])
        h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=1)
        # WENO-Z-5 right-biased: valid for i=1..Nx-4
        h5_neg_int = _wenoz5_right(
            h[1:-1, :-4], h[1:-1, 1:-3], h[1:-1, 2:-2], h[1:-1, 3:-1], h[1:-1, 4:]
        )
        # WENO-Z-3 right-biased fallback at column i=Nx-3
        h3_neg_penultimate = _wenoz3_right(h[1:-1, -3:-2], h[1:-1, -2:-1], h[1:-1, -1:])
        # 1st-order upwind fallback at east boundary column (i=Nx-2)
        h1_neg_last = h[1:-1, -1:]
        h_neg = jnp.concatenate([h5_neg_int, h3_neg_penultimate, h1_neg_last], axis=1)
        h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1], h)
        return out

    def wenoz5_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """5-point WENO-Z north-face flux with sign-dependent boundary fallbacks.

        Positive flow:  fn[j+1/2,i] = WENOZ5(h[j-2..j+2,i]) * v
            for j = 2..Ny-3; WENO-Z-3 fallback at j = 1 and j = Ny-2.
        Negative flow:  fn[j+1/2,i] = WENOZ5_right(h[j-1,i], h[j,i], h[j+1,i], h[j+2,i], h[j+3,i]) * v
            for j = 1..Ny-4; WENO-Z-3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.
        """
        # WENO-Z-5 positive: valid for j=2..Ny-3
        h5_pos_int = _wenoz5(
            h[:-4, 1:-1], h[1:-3, 1:-1], h[2:-2, 1:-1], h[3:-1, 1:-1], h[4:, 1:-1]
        )
        # WENO-Z-3 fallback at first interior row (j=1)
        h3_pos_first = _wenoz3(h[0:1, 1:-1], h[1:2, 1:-1], h[2:3, 1:-1])
        # WENO-Z-3 fallback at last interior row (j=Ny-2)
        h3_pos_last = _wenoz3(h[-3:-2, 1:-1], h[-2:-1, 1:-1], h[-1:, 1:-1])
        h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=0)
        # WENO-Z-5 right-biased: valid for j=1..Ny-4
        h5_neg_int = _wenoz5_right(
            h[:-4, 1:-1], h[1:-3, 1:-1], h[2:-2, 1:-1], h[3:-1, 1:-1], h[4:, 1:-1]
        )
        # WENO-Z-3 right-biased fallback at row j=Ny-3
        h3_neg_penultimate = _wenoz3_right(h[-3:-2, 1:-1], h[-2:-1, 1:-1], h[-1:, 1:-1])
        # 1st-order upwind fallback at north boundary row (j=Ny-2)
        h1_neg_last = h[-1:, 1:-1]
        h_neg = jnp.concatenate([h5_neg_int, h3_neg_penultimate, h1_neg_last], axis=0)
        h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1], h)
        return out

    def weno7_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """7-point WENO east-face flux with hierarchical boundary fallbacks."""
        return _weno_flux_axis_2d_x(h, u, order=7)

    def weno7_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """7-point WENO north-face flux with hierarchical boundary fallbacks."""
        return _weno_flux_axis_2d_y(h, v, order=7)

    def weno9_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """9-point WENO east-face flux with hierarchical boundary fallbacks."""
        return _weno_flux_axis_2d_x(h, u, order=9)

    def weno9_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
    ) -> Float[Array, "Ny Nx"]:
        """9-point WENO north-face flux with hierarchical boundary fallbacks."""
        return _weno_flux_axis_2d_y(h, v, order=9)

    def tvd_x(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
        limiter: str = "minmod",
    ) -> Float[Array, "Ny Nx"]:
        """TVD east-face flux using a flux limiter.

        Positive flow:
            r         = (h[j,i] − h[j,i−1]) / (h[j,i+1] − h[j,i])
            h_L[j,i+½] = h[j,i]   + ½ φ(r) (h[j,i+1] − h[j,i])

        Negative flow:
            r         = (h[j,i+1] − h[j,i+2]) / (h[j,i] − h[j,i+1])
            h_R[j,i+½] = h[j,i+1] + ½ φ(r) (h[j,i]   − h[j,i+1])
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Scalar at T-points.
        u : Float[Array, "Ny Nx"]
            x-velocity at U-points.
        limiter : str
            Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
            or ``'mc'``.

        Returns
        -------
        Float[Array, "Ny Nx"]
            East-face flux with zero ghost ring.
        """
        phi = _get_limiter(limiter)
        # Positive flow: h_face = h[j,i] + 0.5*phi(r)*(h[j,i+1]-h[j,i])
        # r = (h[j,i]-h[j,i-1]) / (h[j,i+1]-h[j,i])
        diff_fwd = h[1:-1, 2:] - h[1:-1, 1:-1]  # h[j,i+1] - h[j,i]
        diff_bwd = h[1:-1, 1:-1] - h[1:-1, :-2]  # h[j,i] - h[j,i-1]
        r_pos = diff_bwd / (diff_fwd + _TVD_EPS)
        h_pos = h[1:-1, 1:-1] + 0.5 * phi(r_pos) * diff_fwd
        # Negative flow: h_face = h[j,i+1] + 0.5*phi(r)*(h[j,i]-h[j,i+1])
        # r = (h[j,i+1]-h[j,i+2]) / (h[j,i]-h[j,i+1])  for i=1..Nx-3
        diff_neg = h[1:-1, 1:-2] - h[1:-1, 2:-1]  # h[j,i] - h[j,i+1]
        diff_neg2 = h[1:-1, 2:-1] - h[1:-1, 3:]  # h[j,i+1] - h[j,i+2]
        r_neg_int = diff_neg2 / (diff_neg + _TVD_EPS)
        h_neg_interior = h[1:-1, 2:-1] + 0.5 * phi(r_neg_int) * diff_neg
        # 1st-order upwind fallback at east boundary column (i=Nx-2)
        h_neg_boundary = h[1:-1, -1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
        h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1], h)
        return out

    def tvd_y(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
        limiter: str = "minmod",
    ) -> Float[Array, "Ny Nx"]:
        """TVD north-face flux using a flux limiter.

        Positive flow:
            r         = (h[j,i] − h[j−1,i]) / (h[j+1,i] − h[j,i])
            h_L[j+½,i] = h[j,i]   + ½ φ(r) (h[j+1,i] − h[j,i])

        Negative flow:
            r         = (h[j+1,i] − h[j+2,i]) / (h[j,i] − h[j+1,i])
            h_R[j+½,i] = h[j+1,i] + ½ φ(r) (h[j,i]   − h[j+1,i])
        Falls back to 1st-order upwind on north boundary where j+2 unavailable.

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Scalar at T-points.
        v : Float[Array, "Ny Nx"]
            y-velocity at V-points.
        limiter : str
            Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
            or ``'mc'``.

        Returns
        -------
        Float[Array, "Ny Nx"]
            North-face flux with zero ghost ring.
        """
        phi = _get_limiter(limiter)
        # Positive flow: h_face = h[j,i] + 0.5*phi(r)*(h[j+1,i]-h[j,i])
        # r = (h[j,i]-h[j-1,i]) / (h[j+1,i]-h[j,i])
        diff_fwd = h[2:, 1:-1] - h[1:-1, 1:-1]  # h[j+1,i] - h[j,i]
        diff_bwd = h[1:-1, 1:-1] - h[:-2, 1:-1]  # h[j,i] - h[j-1,i]
        r_pos = diff_bwd / (diff_fwd + _TVD_EPS)
        h_pos = h[1:-1, 1:-1] + 0.5 * phi(r_pos) * diff_fwd
        # Negative flow: h_face = h[j+1,i] + 0.5*phi(r)*(h[j,i]-h[j+1,i])
        # r = (h[j+1,i]-h[j+2,i]) / (h[j,i]-h[j+1,i])  for j=1..Ny-3
        diff_neg = h[1:-2, 1:-1] - h[2:-1, 1:-1]  # h[j,i] - h[j+1,i]
        diff_neg2 = h[2:-1, 1:-1] - h[3:, 1:-1]  # h[j+1,i] - h[j+2,i]
        r_neg_int = diff_neg2 / (diff_neg + _TVD_EPS)
        h_neg_interior = h[2:-1, 1:-1] + 0.5 * phi(r_neg_int) * diff_neg
        # 1st-order upwind fallback at north boundary row (j=Ny-2)
        h_neg_boundary = h[-1:, 1:-1]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=0)
        h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1], h)
        return out

    def tvd_x_masked(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
        mask: ArakawaCGridMask,
        limiter: str = "minmod",
    ) -> Float[Array, "Ny Nx"]:
        """TVD east-face flux with mask-aware stencil selection.

        Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
        two-tier hierarchy (upwind1 / TVD).

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Cell-centre tracer field.
        u : Float[Array, "Ny Nx"]
            East-face velocity.
        mask : ArakawaCGridMask
            Arakawa C-grid mask providing stencil-capability information.
        limiter : str
            Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
            or ``'mc'``.

        Returns
        -------
        Float[Array, "Ny Nx"]
            East-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4))
        rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
            2: self.upwind1_x,
            4: lambda q, vel: self.tvd_x(q, vel, limiter=limiter),
        }
        return upwind_flux(h, u, dim=1, rec_funcs=rec_funcs, mask_hierarchy=amasks)

    def tvd_y_masked(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
        mask: ArakawaCGridMask,
        limiter: str = "minmod",
    ) -> Float[Array, "Ny Nx"]:
        """TVD north-face flux with mask-aware stencil selection.

        Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
        two-tier hierarchy (upwind1 / TVD).

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Cell-centre tracer field.
        v : Float[Array, "Ny Nx"]
            North-face velocity.
        mask : ArakawaCGridMask
            Arakawa C-grid mask providing stencil-capability information.
        limiter : str
            Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
            or ``'mc'``.

        Returns
        -------
        Float[Array, "Ny Nx"]
            North-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4))
        rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
            2: self.upwind1_y,
            4: lambda q, vel: self.tvd_y(q, vel, limiter=limiter),
        }
        return upwind_flux(h, v, dim=0, rec_funcs=rec_funcs, mask_hierarchy=amasks)

    def weno5_x_masked(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
        mask: ArakawaCGridMask,
    ) -> Float[Array, "Ny Nx"]:
        """5-point WENO east-face flux with mask-aware adaptive stencil selection.

        Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
        three-tier hierarchy (upwind1 / WENO3 / WENO5).

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Cell-centre tracer field.
        u : Float[Array, "Ny Nx"]
            East-face velocity.
        mask : ArakawaCGridMask
            Arakawa C-grid mask providing stencil-capability information.

        Returns
        -------
        Float[Array, "Ny Nx"]
            East-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4, 6))
        rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
            2: self.upwind1_x,
            4: self.weno3_x,
            6: self.weno5_x,
        }
        return upwind_flux(h, u, dim=1, rec_funcs=rec_funcs, mask_hierarchy=amasks)

    def weno5_y_masked(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
        mask: ArakawaCGridMask,
    ) -> Float[Array, "Ny Nx"]:
        """5-point WENO north-face flux with mask-aware adaptive stencil selection.

        Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
        three-tier hierarchy (upwind1 / WENO3 / WENO5).

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Cell-centre tracer field.
        v : Float[Array, "Ny Nx"]
            North-face velocity.
        mask : ArakawaCGridMask
            Arakawa C-grid mask providing stencil-capability information.

        Returns
        -------
        Float[Array, "Ny Nx"]
            North-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4, 6))
        rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
            2: self.upwind1_y,
            4: self.weno3_y,
            6: self.weno5_y,
        }
        return upwind_flux(h, v, dim=0, rec_funcs=rec_funcs, mask_hierarchy=amasks)

    def wenoz5_x_masked(
        self,
        h: Float[Array, "Ny Nx"],
        u: Float[Array, "Ny Nx"],
        mask: ArakawaCGridMask,
    ) -> Float[Array, "Ny Nx"]:
        """5-point WENO-Z east-face flux with mask-aware adaptive stencil selection.

        Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
        three-tier hierarchy (upwind1 / WENO-Z-3 / WENO-Z-5).

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Cell-centre tracer field.
        u : Float[Array, "Ny Nx"]
            East-face velocity.
        mask : ArakawaCGridMask
            Arakawa C-grid mask providing stencil-capability information.

        Returns
        -------
        Float[Array, "Ny Nx"]
            East-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4, 6))
        rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
            2: self.upwind1_x,
            4: self.wenoz3_x,
            6: self.wenoz5_x,
        }
        return upwind_flux(h, u, dim=1, rec_funcs=rec_funcs, mask_hierarchy=amasks)

    def wenoz5_y_masked(
        self,
        h: Float[Array, "Ny Nx"],
        v: Float[Array, "Ny Nx"],
        mask: ArakawaCGridMask,
    ) -> Float[Array, "Ny Nx"]:
        """5-point WENO-Z north-face flux with mask-aware adaptive stencil selection.

        Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
        three-tier hierarchy (upwind1 / WENO-Z-3 / WENO-Z-5).

        Parameters
        ----------
        h : Float[Array, "Ny Nx"]
            Cell-centre tracer field.
        v : Float[Array, "Ny Nx"]
            North-face velocity.
        mask : ArakawaCGridMask
            Arakawa C-grid mask providing stencil-capability information.

        Returns
        -------
        Float[Array, "Ny Nx"]
            North-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4, 6))
        rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
            2: self.upwind1_y,
            4: self.wenoz3_y,
            6: self.wenoz5_y,
        }
        return upwind_flux(h, v, dim=0, rec_funcs=rec_funcs, mask_hierarchy=amasks)

naive_x(h, u)

Naive (centred) east-face flux.

fe[j, i+1/2] = 1/2 * (h[j, i] + h[j, i+1]) * u[j, i+1/2]

Source code in finitevolx/_src/advection/reconstruction.py 
578
579
580
581
582
583
584
585
586
587
588
def naive_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """Naive (centred) east-face flux.

    fe[j, i+1/2] = 1/2 * (h[j, i] + h[j, i+1]) * u[j, i+1/2]
    """
    out = interior(0.5 * (h[1:-1, 1:-1] + h[1:-1, 2:]) * u[1:-1, 1:-1], h)
    return out

naive_y(h, v)

Naive (centred) north-face flux.

fn[j+1/2, i] = 1/2 * (h[j, i] + h[j+1, i]) * v[j+1/2, i]

Source code in finitevolx/_src/advection/reconstruction.py 
590
591
592
593
594
595
596
597
598
599
600
def naive_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """Naive (centred) north-face flux.

    fn[j+1/2, i] = 1/2 * (h[j, i] + h[j+1, i]) * v[j+1/2, i]
    """
    out = interior(0.5 * (h[1:-1, 1:-1] + h[2:, 1:-1]) * v[1:-1, 1:-1], h)
    return out

tvd_x(h, u, limiter='minmod')

TVD east-face flux using a flux limiter.

Positive flow: r = (h[j,i] − h[j,i−1]) / (h[j,i+1] − h[j,i]) h_L[j,i+½] = h[j,i] + ½ φ(r) (h[j,i+1] − h[j,i])

Negative flow: r = (h[j,i+1] − h[j,i+2]) / (h[j,i] − h[j,i+1]) h_R[j,i+½] = h[j,i+1] + ½ φ(r) (h[j,i] − h[j,i+1]) Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Scalar at T-points.

 required
u Float[Array, 'Ny Nx']

x-velocity at U-points.

 required
limiter str

Flux limiter name: 'minmod', 'van_leer', 'superbee', or 'mc'.

 'minmod'

Returns:

Type Description
Float[Array, 'Ny Nx']

East-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
 996
 997
 998
 999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
def tvd_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
    limiter: str = "minmod",
) -> Float[Array, "Ny Nx"]:
    """TVD east-face flux using a flux limiter.

    Positive flow:
        r         = (h[j,i] − h[j,i−1]) / (h[j,i+1] − h[j,i])
        h_L[j,i+½] = h[j,i]   + ½ φ(r) (h[j,i+1] − h[j,i])

    Negative flow:
        r         = (h[j,i+1] − h[j,i+2]) / (h[j,i] − h[j,i+1])
        h_R[j,i+½] = h[j,i+1] + ½ φ(r) (h[j,i]   − h[j,i+1])
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Scalar at T-points.
    u : Float[Array, "Ny Nx"]
        x-velocity at U-points.
    limiter : str
        Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
        or ``'mc'``.

    Returns
    -------
    Float[Array, "Ny Nx"]
        East-face flux with zero ghost ring.
    """
    phi = _get_limiter(limiter)
    # Positive flow: h_face = h[j,i] + 0.5*phi(r)*(h[j,i+1]-h[j,i])
    # r = (h[j,i]-h[j,i-1]) / (h[j,i+1]-h[j,i])
    diff_fwd = h[1:-1, 2:] - h[1:-1, 1:-1]  # h[j,i+1] - h[j,i]
    diff_bwd = h[1:-1, 1:-1] - h[1:-1, :-2]  # h[j,i] - h[j,i-1]
    r_pos = diff_bwd / (diff_fwd + _TVD_EPS)
    h_pos = h[1:-1, 1:-1] + 0.5 * phi(r_pos) * diff_fwd
    # Negative flow: h_face = h[j,i+1] + 0.5*phi(r)*(h[j,i]-h[j,i+1])
    # r = (h[j,i+1]-h[j,i+2]) / (h[j,i]-h[j,i+1])  for i=1..Nx-3
    diff_neg = h[1:-1, 1:-2] - h[1:-1, 2:-1]  # h[j,i] - h[j,i+1]
    diff_neg2 = h[1:-1, 2:-1] - h[1:-1, 3:]  # h[j,i+1] - h[j,i+2]
    r_neg_int = diff_neg2 / (diff_neg + _TVD_EPS)
    h_neg_interior = h[1:-1, 2:-1] + 0.5 * phi(r_neg_int) * diff_neg
    # 1st-order upwind fallback at east boundary column (i=Nx-2)
    h_neg_boundary = h[1:-1, -1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
    h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1], h)
    return out

tvd_x_masked(h, u, mask, limiter='minmod')

TVD east-face flux with mask-aware stencil selection.

Delegates to :func:~finitevolx._src.advection.flux.upwind_flux with a two-tier hierarchy (upwind1 / TVD).

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Cell-centre tracer field.

 required
u Float[Array, 'Ny Nx']

East-face velocity.

 required
mask ArakawaCGridMask

Arakawa C-grid mask providing stencil-capability information.

 required
limiter str

Flux limiter name: 'minmod', 'van_leer', 'superbee', or 'mc'.

 'minmod'

Returns:

Type Description
Float[Array, 'Ny Nx']

East-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
def tvd_x_masked(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
    mask: ArakawaCGridMask,
    limiter: str = "minmod",
) -> Float[Array, "Ny Nx"]:
    """TVD east-face flux with mask-aware stencil selection.

    Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
    two-tier hierarchy (upwind1 / TVD).

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Cell-centre tracer field.
    u : Float[Array, "Ny Nx"]
        East-face velocity.
    mask : ArakawaCGridMask
        Arakawa C-grid mask providing stencil-capability information.
    limiter : str
        Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
        or ``'mc'``.

    Returns
    -------
    Float[Array, "Ny Nx"]
        East-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4))
    rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
        2: self.upwind1_x,
        4: lambda q, vel: self.tvd_x(q, vel, limiter=limiter),
    }
    return upwind_flux(h, u, dim=1, rec_funcs=rec_funcs, mask_hierarchy=amasks)

tvd_y(h, v, limiter='minmod')

TVD north-face flux using a flux limiter.

Positive flow: r = (h[j,i] − h[j−1,i]) / (h[j+1,i] − h[j,i]) h_L[j+½,i] = h[j,i] + ½ φ(r) (h[j+1,i] − h[j,i])

Negative flow: r = (h[j+1,i] − h[j+2,i]) / (h[j,i] − h[j+1,i]) h_R[j+½,i] = h[j+1,i] + ½ φ(r) (h[j,i] − h[j+1,i]) Falls back to 1st-order upwind on north boundary where j+2 unavailable.

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Scalar at T-points.

 required
v Float[Array, 'Ny Nx']

y-velocity at V-points.

 required
limiter str

Flux limiter name: 'minmod', 'van_leer', 'superbee', or 'mc'.

 'minmod'

Returns:

Type Description
Float[Array, 'Ny Nx']

North-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
def tvd_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    limiter: str = "minmod",
) -> Float[Array, "Ny Nx"]:
    """TVD north-face flux using a flux limiter.

    Positive flow:
        r         = (h[j,i] − h[j−1,i]) / (h[j+1,i] − h[j,i])
        h_L[j+½,i] = h[j,i]   + ½ φ(r) (h[j+1,i] − h[j,i])

    Negative flow:
        r         = (h[j+1,i] − h[j+2,i]) / (h[j,i] − h[j+1,i])
        h_R[j+½,i] = h[j+1,i] + ½ φ(r) (h[j,i]   − h[j+1,i])
    Falls back to 1st-order upwind on north boundary where j+2 unavailable.

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Scalar at T-points.
    v : Float[Array, "Ny Nx"]
        y-velocity at V-points.
    limiter : str
        Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
        or ``'mc'``.

    Returns
    -------
    Float[Array, "Ny Nx"]
        North-face flux with zero ghost ring.
    """
    phi = _get_limiter(limiter)
    # Positive flow: h_face = h[j,i] + 0.5*phi(r)*(h[j+1,i]-h[j,i])
    # r = (h[j,i]-h[j-1,i]) / (h[j+1,i]-h[j,i])
    diff_fwd = h[2:, 1:-1] - h[1:-1, 1:-1]  # h[j+1,i] - h[j,i]
    diff_bwd = h[1:-1, 1:-1] - h[:-2, 1:-1]  # h[j,i] - h[j-1,i]
    r_pos = diff_bwd / (diff_fwd + _TVD_EPS)
    h_pos = h[1:-1, 1:-1] + 0.5 * phi(r_pos) * diff_fwd
    # Negative flow: h_face = h[j+1,i] + 0.5*phi(r)*(h[j,i]-h[j+1,i])
    # r = (h[j+1,i]-h[j+2,i]) / (h[j,i]-h[j+1,i])  for j=1..Ny-3
    diff_neg = h[1:-2, 1:-1] - h[2:-1, 1:-1]  # h[j,i] - h[j+1,i]
    diff_neg2 = h[2:-1, 1:-1] - h[3:, 1:-1]  # h[j+1,i] - h[j+2,i]
    r_neg_int = diff_neg2 / (diff_neg + _TVD_EPS)
    h_neg_interior = h[2:-1, 1:-1] + 0.5 * phi(r_neg_int) * diff_neg
    # 1st-order upwind fallback at north boundary row (j=Ny-2)
    h_neg_boundary = h[-1:, 1:-1]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=0)
    h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1], h)
    return out

tvd_y_masked(h, v, mask, limiter='minmod')

TVD north-face flux with mask-aware stencil selection.

Delegates to :func:~finitevolx._src.advection.flux.upwind_flux with a two-tier hierarchy (upwind1 / TVD).

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Cell-centre tracer field.

 required
v Float[Array, 'Ny Nx']

North-face velocity.

 required
mask ArakawaCGridMask

Arakawa C-grid mask providing stencil-capability information.

 required
limiter str

Flux limiter name: 'minmod', 'van_leer', 'superbee', or 'mc'.

 'minmod'

Returns:

Type Description
Float[Array, 'Ny Nx']

North-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
def tvd_y_masked(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    mask: ArakawaCGridMask,
    limiter: str = "minmod",
) -> Float[Array, "Ny Nx"]:
    """TVD north-face flux with mask-aware stencil selection.

    Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
    two-tier hierarchy (upwind1 / TVD).

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Cell-centre tracer field.
    v : Float[Array, "Ny Nx"]
        North-face velocity.
    mask : ArakawaCGridMask
        Arakawa C-grid mask providing stencil-capability information.
    limiter : str
        Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
        or ``'mc'``.

    Returns
    -------
    Float[Array, "Ny Nx"]
        North-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4))
    rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
        2: self.upwind1_y,
        4: lambda q, vel: self.tvd_y(q, vel, limiter=limiter),
    }
    return upwind_flux(h, v, dim=0, rec_funcs=rec_funcs, mask_hierarchy=amasks)

upwind1_x(h, u)

1st-order upwind east-face flux.

fe[j, i+1/2] = h[j, i] * u if u >= 0 = h[j, i+1] * u otherwise

Source code in finitevolx/_src/advection/reconstruction.py 
602
603
604
605
606
607
608
609
610
611
612
613
614
615
def upwind1_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """1st-order upwind east-face flux.

    fe[j, i+1/2] = h[j, i]   * u  if u >= 0
                 = h[j, i+1] * u  otherwise
    """
    # fe[j, i+1/2] = upwind1(h, u) * u[j, i+1/2]
    h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h[1:-1, 1:-1], h[1:-1, 2:])
    out = interior(h_face * u[1:-1, 1:-1], h)
    return out

upwind1_y(h, v)

1st-order upwind north-face flux.

fn[j+1/2, i] = h[j, i] * v if v >= 0 = h[j+1, i] * v otherwise

Source code in finitevolx/_src/advection/reconstruction.py 
617
618
619
620
621
622
623
624
625
626
627
628
629
630
def upwind1_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """1st-order upwind north-face flux.

    fn[j+1/2, i] = h[j, i]   * v  if v >= 0
                 = h[j+1, i] * v  otherwise
    """
    # fn[j+1/2, i] = upwind1(h, v) * v[j+1/2, i]
    h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h[1:-1, 1:-1], h[2:, 1:-1])
    out = interior(h_face * v[1:-1, 1:-1], h)
    return out

upwind2_x(h, u)

2nd-order upwind east-face flux with boundary fallback.

Positive: h_face = 3/2h[j,i] - 1/2h[j,i-1] Negative: h_face = 3/2h[j,i+1] - 1/2h[j,i+2] Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
def upwind2_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """2nd-order upwind east-face flux with boundary fallback.

    Positive: h_face = 3/2*h[j,i] - 1/2*h[j,i-1]
    Negative: h_face = 3/2*h[j,i+1] - 1/2*h[j,i+2]
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.
    """
    # 2nd-order positive stencil (uses h[j,i-1])
    h_pos = 1.5 * h[1:-1, 1:-1] - 0.5 * h[1:-1, :-2]
    # 2nd-order negative stencil (uses h[j,i+2])
    # Use 2nd-order where i+2 is available (all except last interior column)
    h_neg_interior = 1.5 * h[1:-1, 2:-1] - 0.5 * h[1:-1, 3:]
    # Use 1st-order upwind on east boundary column (h_face = h[j,i+1])
    h_neg_boundary = h[1:-1, -1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
    h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1], h)
    return out

upwind2_y(h, v)

2nd-order upwind north-face flux with boundary fallback.

Positive: h_face = 3/2h[j,i] - 1/2h[j-1,i] Negative: h_face = 3/2h[j+1,i] - 1/2h[j+2,i] Falls back to 1st-order upwind on north boundary where j+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
def upwind2_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """2nd-order upwind north-face flux with boundary fallback.

    Positive: h_face = 3/2*h[j,i] - 1/2*h[j-1,i]
    Negative: h_face = 3/2*h[j+1,i] - 1/2*h[j+2,i]
    Falls back to 1st-order upwind on north boundary where j+2 unavailable.
    """
    # 2nd-order positive stencil (uses h[j-1,i])
    h_pos = 1.5 * h[1:-1, 1:-1] - 0.5 * h[:-2, 1:-1]
    # 2nd-order negative stencil (uses h[j+2,i])
    # Use 2nd-order where j+2 is available (all except last interior row)
    h_neg_interior = 1.5 * h[2:-1, 1:-1] - 0.5 * h[3:, 1:-1]
    # Use 1st-order upwind on north boundary row (h_face = h[j+1,i])
    h_neg_boundary = h[-1:, 1:-1]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=0)
    h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1], h)
    return out

upwind3_x(h, u)

3rd-order upwind east-face flux with boundary fallback.

Positive: h_face = -1/6h[j,i-1] + 5/6h[j,i] + 1/3h[j,i+1] Negative: h_face = 1/3h[j,i] + 5/6h[j,i+1] - 1/6h[j,i+2] Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
def upwind3_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """3rd-order upwind east-face flux with boundary fallback.

    Positive: h_face = -1/6*h[j,i-1] + 5/6*h[j,i]   + 1/3*h[j,i+1]
    Negative: h_face =  1/3*h[j,i]   + 5/6*h[j,i+1] - 1/6*h[j,i+2]
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.
    """
    # 3rd-order positive stencil (valid for all interior faces)
    h_pos = (
        -1.0 / 6.0 * h[1:-1, :-2]  # h[j, i-1]
        + 5.0 / 6.0 * h[1:-1, 1:-1]  # h[j, i  ]
        + 1.0 / 3.0 * h[1:-1, 2:]  # h[j, i+1]
    )
    # 3rd-order negative stencil (valid only where h[j,i+2] exists)
    # For all interior columns except the last one
    h_neg_interior = (
        1.0 / 3.0 * h[1:-1, 1:-2]  # h[j, i  ]
        + 5.0 / 6.0 * h[1:-1, 2:-1]  # h[j, i+1]
        - 1.0 / 6.0 * h[1:-1, 3:]  # h[j, i+2]
    )
    # 1st-order upwind fallback at east boundary: h_face = h[j,i+1]
    h_neg_boundary = h[1:-1, -1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
    h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1], h)
    return out

upwind3_y(h, v)

3rd-order upwind north-face flux with boundary fallback.

Positive: h_face = -1/6h[j-1,i] + 5/6h[j,i] + 1/3h[j+1,i] Negative: h_face = 1/3h[j,i] + 5/6h[j+1,i] - 1/6h[j+2,i] Falls back to 1st-order upwind on north boundary where j+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
def upwind3_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """3rd-order upwind north-face flux with boundary fallback.

    Positive: h_face = -1/6*h[j-1,i] + 5/6*h[j,i]   + 1/3*h[j+1,i]
    Negative: h_face =  1/3*h[j,i]   + 5/6*h[j+1,i] - 1/6*h[j+2,i]
    Falls back to 1st-order upwind on north boundary where j+2 unavailable.
    """
    # 3rd-order positive stencil (valid for all interior faces)
    h_pos = (
        -1.0 / 6.0 * h[:-2, 1:-1]  # h[j-1, i]
        + 5.0 / 6.0 * h[1:-1, 1:-1]  # h[j,   i]
        + 1.0 / 3.0 * h[2:, 1:-1]  # h[j+1, i]
    )
    # 3rd-order negative stencil (valid only where h[j+2,i] exists)
    # For all interior rows except the last one
    h_neg_interior = (
        1.0 / 3.0 * h[1:-2, 1:-1]  # h[j,   i]
        + 5.0 / 6.0 * h[2:-1, 1:-1]  # h[j+1, i]
        - 1.0 / 6.0 * h[3:, 1:-1]  # h[j+2, i]
    )
    # 1st-order upwind fallback at north boundary: h_face = h[j+1,i]
    h_neg_boundary = h[-1:, 1:-1]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=0)
    h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1], h)
    return out

weno3_x(h, u)

3-point WENO east-face flux with boundary fallback.

Positive flow: fe[j,i+1/2] = WENO3(h[j,i-1], h[j,i], h[j,i+1]) * u Negative flow: fe[j,i+1/2] = WENO3_right(h[j,i], h[j,i+1], h[j,i+2]) * u Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
def weno3_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """3-point WENO east-face flux with boundary fallback.

    Positive flow:  fe[j,i+1/2] = WENO3(h[j,i-1], h[j,i],   h[j,i+1]) * u
    Negative flow:  fe[j,i+1/2] = WENO3_right(h[j,i], h[j,i+1], h[j,i+2]) * u
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.
    """
    # WENO-3 positive: left-biased, valid for all interior faces
    h_pos = _weno3(h[1:-1, :-2], h[1:-1, 1:-1], h[1:-1, 2:])
    # WENO-3 negative: right-biased, valid for i+2 < Nx
    h_neg_interior = _weno3_right(h[1:-1, 1:-2], h[1:-1, 2:-1], h[1:-1, 3:])
    # 1st-order upwind fallback at east boundary column
    h_neg_boundary = h[1:-1, -1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
    h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1], h)
    return out

weno3_y(h, v)

3-point WENO north-face flux with boundary fallback.

Positive flow: fn[j+1/2,i] = WENO3(h[j-1,i], h[j,i], h[j+1,i]) * v Negative flow: fn[j+1/2,i] = WENO3_right(h[j,i], h[j+1,i], h[j+2,i]) * v Falls back to 1st-order upwind on north boundary where j+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
def weno3_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """3-point WENO north-face flux with boundary fallback.

    Positive flow:  fn[j+1/2,i] = WENO3(h[j-1,i], h[j,i],   h[j+1,i]) * v
    Negative flow:  fn[j+1/2,i] = WENO3_right(h[j,i], h[j+1,i], h[j+2,i]) * v
    Falls back to 1st-order upwind on north boundary where j+2 unavailable.
    """
    # WENO-3 positive: left-biased, valid for all interior faces
    h_pos = _weno3(h[:-2, 1:-1], h[1:-1, 1:-1], h[2:, 1:-1])
    # WENO-3 negative: right-biased, valid for j+2 < Ny
    h_neg_interior = _weno3_right(h[1:-2, 1:-1], h[2:-1, 1:-1], h[3:, 1:-1])
    # 1st-order upwind fallback at north boundary row
    h_neg_boundary = h[-1:, 1:-1]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=0)
    h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1], h)
    return out

weno5_x(h, u)

5-point WENO east-face flux with sign-dependent boundary fallbacks.

Positive flow: fe[j,i+1/2] = WENO5(h[j,i-2..i+2]) * u for i = 2..Nx-3; WENO3 fallback at i = 1 and i = Nx-2. Negative flow: fe[j,i+1/2] = WENO5_right(h[j,i-1], h[j,i], h[j,i+1], h[j,i+2], h[j,i+3]) * u for i = 1..Nx-4; WENO3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.

Source code in finitevolx/_src/advection/reconstruction.py 
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
def weno5_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """5-point WENO east-face flux with sign-dependent boundary fallbacks.

    Positive flow:  fe[j,i+1/2] = WENO5(h[j,i-2..i+2]) * u
        for i = 2..Nx-3; WENO3 fallback at i = 1 and i = Nx-2.
    Negative flow:  fe[j,i+1/2] = WENO5_right(h[j,i-1], h[j,i], h[j,i+1], h[j,i+2], h[j,i+3]) * u
        for i = 1..Nx-4; WENO3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
    """
    # WENO-5 positive: valid for i=2..Nx-3
    h5_pos_int = _weno5(
        h[1:-1, :-4], h[1:-1, 1:-3], h[1:-1, 2:-2], h[1:-1, 3:-1], h[1:-1, 4:]
    )
    # WENO-3 fallback at first interior column (i=1)
    h3_pos_first = _weno3(h[1:-1, 0:1], h[1:-1, 1:2], h[1:-1, 2:3])
    # WENO-3 fallback at last interior column (i=Nx-2)
    h3_pos_last = _weno3(h[1:-1, -3:-2], h[1:-1, -2:-1], h[1:-1, -1:])
    h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=1)
    # WENO-5 right-biased: valid for i=1..Nx-4
    h5_neg_int = _weno5_right(
        h[1:-1, :-4], h[1:-1, 1:-3], h[1:-1, 2:-2], h[1:-1, 3:-1], h[1:-1, 4:]
    )
    # WENO-3 right-biased fallback at column i=Nx-3
    h3_neg_penultimate = _weno3_right(h[1:-1, -3:-2], h[1:-1, -2:-1], h[1:-1, -1:])
    # 1st-order upwind fallback at east boundary column (i=Nx-2)
    h1_neg_last = h[1:-1, -1:]
    h_neg = jnp.concatenate([h5_neg_int, h3_neg_penultimate, h1_neg_last], axis=1)
    h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1], h)
    return out

weno5_x_masked(h, u, mask)

5-point WENO east-face flux with mask-aware adaptive stencil selection.

Delegates to :func:~finitevolx._src.advection.flux.upwind_flux with a three-tier hierarchy (upwind1 / WENO3 / WENO5).

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Cell-centre tracer field.

 required
u Float[Array, 'Ny Nx']

East-face velocity.

 required
mask ArakawaCGridMask

Arakawa C-grid mask providing stencil-capability information.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

East-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
def weno5_x_masked(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
    mask: ArakawaCGridMask,
) -> Float[Array, "Ny Nx"]:
    """5-point WENO east-face flux with mask-aware adaptive stencil selection.

    Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
    three-tier hierarchy (upwind1 / WENO3 / WENO5).

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Cell-centre tracer field.
    u : Float[Array, "Ny Nx"]
        East-face velocity.
    mask : ArakawaCGridMask
        Arakawa C-grid mask providing stencil-capability information.

    Returns
    -------
    Float[Array, "Ny Nx"]
        East-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4, 6))
    rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
        2: self.upwind1_x,
        4: self.weno3_x,
        6: self.weno5_x,
    }
    return upwind_flux(h, u, dim=1, rec_funcs=rec_funcs, mask_hierarchy=amasks)

weno5_y(h, v)

5-point WENO north-face flux with sign-dependent boundary fallbacks.

Positive flow: fn[j+1/2,i] = WENO5(h[j-2..j+2,i]) * v for j = 2..Ny-3; WENO3 fallback at j = 1 and j = Ny-2. Negative flow: fn[j+1/2,i] = WENO5_right(h[j-1,i], h[j,i], h[j+1,i], h[j+2,i], h[j+3,i]) * v for j = 1..Ny-4; WENO3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.

Source code in finitevolx/_src/advection/reconstruction.py 
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
def weno5_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """5-point WENO north-face flux with sign-dependent boundary fallbacks.

    Positive flow:  fn[j+1/2,i] = WENO5(h[j-2..j+2,i]) * v
        for j = 2..Ny-3; WENO3 fallback at j = 1 and j = Ny-2.
    Negative flow:  fn[j+1/2,i] = WENO5_right(h[j-1,i], h[j,i], h[j+1,i], h[j+2,i], h[j+3,i]) * v
        for j = 1..Ny-4; WENO3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.
    """
    # WENO-5 positive: valid for j=2..Ny-3
    h5_pos_int = _weno5(
        h[:-4, 1:-1], h[1:-3, 1:-1], h[2:-2, 1:-1], h[3:-1, 1:-1], h[4:, 1:-1]
    )
    # WENO-3 fallback at first interior row (j=1)
    h3_pos_first = _weno3(h[0:1, 1:-1], h[1:2, 1:-1], h[2:3, 1:-1])
    # WENO-3 fallback at last interior row (j=Ny-2)
    h3_pos_last = _weno3(h[-3:-2, 1:-1], h[-2:-1, 1:-1], h[-1:, 1:-1])
    h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=0)
    # WENO-5 right-biased: valid for j=1..Ny-4
    h5_neg_int = _weno5_right(
        h[:-4, 1:-1], h[1:-3, 1:-1], h[2:-2, 1:-1], h[3:-1, 1:-1], h[4:, 1:-1]
    )
    # WENO-3 right-biased fallback at row j=Ny-3
    h3_neg_penultimate = _weno3_right(h[-3:-2, 1:-1], h[-2:-1, 1:-1], h[-1:, 1:-1])
    # 1st-order upwind fallback at north boundary row (j=Ny-2)
    h1_neg_last = h[-1:, 1:-1]
    h_neg = jnp.concatenate([h5_neg_int, h3_neg_penultimate, h1_neg_last], axis=0)
    h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1], h)
    return out

weno5_y_masked(h, v, mask)

5-point WENO north-face flux with mask-aware adaptive stencil selection.

Delegates to :func:~finitevolx._src.advection.flux.upwind_flux with a three-tier hierarchy (upwind1 / WENO3 / WENO5).

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Cell-centre tracer field.

 required
v Float[Array, 'Ny Nx']

North-face velocity.

 required
mask ArakawaCGridMask

Arakawa C-grid mask providing stencil-capability information.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

North-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
def weno5_y_masked(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    mask: ArakawaCGridMask,
) -> Float[Array, "Ny Nx"]:
    """5-point WENO north-face flux with mask-aware adaptive stencil selection.

    Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
    three-tier hierarchy (upwind1 / WENO3 / WENO5).

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Cell-centre tracer field.
    v : Float[Array, "Ny Nx"]
        North-face velocity.
    mask : ArakawaCGridMask
        Arakawa C-grid mask providing stencil-capability information.

    Returns
    -------
    Float[Array, "Ny Nx"]
        North-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4, 6))
    rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
        2: self.upwind1_y,
        4: self.weno3_y,
        6: self.weno5_y,
    }
    return upwind_flux(h, v, dim=0, rec_funcs=rec_funcs, mask_hierarchy=amasks)

weno7_x(h, u)

7-point WENO east-face flux with hierarchical boundary fallbacks.

Source code in finitevolx/_src/advection/reconstruction.py 
964
965
966
967
968
969
970
def weno7_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """7-point WENO east-face flux with hierarchical boundary fallbacks."""
    return _weno_flux_axis_2d_x(h, u, order=7)

weno7_y(h, v)

7-point WENO north-face flux with hierarchical boundary fallbacks.

Source code in finitevolx/_src/advection/reconstruction.py 
972
973
974
975
976
977
978
def weno7_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """7-point WENO north-face flux with hierarchical boundary fallbacks."""
    return _weno_flux_axis_2d_y(h, v, order=7)

weno9_x(h, u)

9-point WENO east-face flux with hierarchical boundary fallbacks.

Source code in finitevolx/_src/advection/reconstruction.py 
980
981
982
983
984
985
986
def weno9_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """9-point WENO east-face flux with hierarchical boundary fallbacks."""
    return _weno_flux_axis_2d_x(h, u, order=9)

weno9_y(h, v)

9-point WENO north-face flux with hierarchical boundary fallbacks.

Source code in finitevolx/_src/advection/reconstruction.py 
988
989
990
991
992
993
994
def weno9_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """9-point WENO north-face flux with hierarchical boundary fallbacks."""
    return _weno_flux_axis_2d_y(h, v, order=9)

wenoz3_x(h, u)

3-point WENO-Z east-face flux with boundary fallback.

Positive flow: fe[j,i+1/2] = WENOZ3(h[j,i-1], h[j,i], h[j,i+1]) * u Negative flow: fe[j,i+1/2] = WENOZ3_right(h[j,i], h[j,i+1], h[j,i+2]) * u Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
def wenoz3_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """3-point WENO-Z east-face flux with boundary fallback.

    Positive flow:  fe[j,i+1/2] = WENOZ3(h[j,i-1], h[j,i],   h[j,i+1]) * u
    Negative flow:  fe[j,i+1/2] = WENOZ3_right(h[j,i], h[j,i+1], h[j,i+2]) * u
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.
    """
    # WENO-Z-3 positive: left-biased, valid for all interior faces
    h_pos = _wenoz3(h[1:-1, :-2], h[1:-1, 1:-1], h[1:-1, 2:])
    # WENO-Z-3 negative: right-biased, valid for i+2 < Nx
    h_neg_interior = _wenoz3_right(h[1:-1, 1:-2], h[1:-1, 2:-1], h[1:-1, 3:])
    # 1st-order upwind fallback at east boundary column
    h_neg_boundary = h[1:-1, -1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
    h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1], h)
    return out

wenoz3_y(h, v)

3-point WENO-Z north-face flux with boundary fallback.

Positive flow: fn[j+1/2,i] = WENOZ3(h[j-1,i], h[j,i], h[j+1,i]) * v Negative flow: fn[j+1/2,i] = WENOZ3_right(h[j,i], h[j+1,i], h[j+2,i]) * v Falls back to 1st-order upwind on north boundary where j+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
def wenoz3_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """3-point WENO-Z north-face flux with boundary fallback.

    Positive flow:  fn[j+1/2,i] = WENOZ3(h[j-1,i], h[j,i],   h[j+1,i]) * v
    Negative flow:  fn[j+1/2,i] = WENOZ3_right(h[j,i], h[j+1,i], h[j+2,i]) * v
    Falls back to 1st-order upwind on north boundary where j+2 unavailable.
    """
    # WENO-Z-3 positive: left-biased, valid for all interior faces
    h_pos = _wenoz3(h[:-2, 1:-1], h[1:-1, 1:-1], h[2:, 1:-1])
    # WENO-Z-3 negative: right-biased, valid for j+2 < Ny
    h_neg_interior = _wenoz3_right(h[1:-2, 1:-1], h[2:-1, 1:-1], h[3:, 1:-1])
    # 1st-order upwind fallback at north boundary row
    h_neg_boundary = h[-1:, 1:-1]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=0)
    h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1], h)
    return out

wenoz5_x(h, u)

5-point WENO-Z east-face flux with sign-dependent boundary fallbacks.

Positive flow: fe[j,i+1/2] = WENOZ5(h[j,i-2..i+2]) * u for i = 2..Nx-3; WENO-Z-3 fallback at i = 1 and i = Nx-2. Negative flow: fe[j,i+1/2] = WENOZ5_right(h[j,i-1], h[j,i], h[j,i+1], h[j,i+2], h[j,i+3]) * u for i = 1..Nx-4; WENO-Z-3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.

Source code in finitevolx/_src/advection/reconstruction.py 
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
def wenoz5_x(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """5-point WENO-Z east-face flux with sign-dependent boundary fallbacks.

    Positive flow:  fe[j,i+1/2] = WENOZ5(h[j,i-2..i+2]) * u
        for i = 2..Nx-3; WENO-Z-3 fallback at i = 1 and i = Nx-2.
    Negative flow:  fe[j,i+1/2] = WENOZ5_right(h[j,i-1], h[j,i], h[j,i+1], h[j,i+2], h[j,i+3]) * u
        for i = 1..Nx-4; WENO-Z-3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
    """
    # WENO-Z-5 positive: valid for i=2..Nx-3
    h5_pos_int = _wenoz5(
        h[1:-1, :-4], h[1:-1, 1:-3], h[1:-1, 2:-2], h[1:-1, 3:-1], h[1:-1, 4:]
    )
    # WENO-Z-3 fallback at first interior column (i=1)
    h3_pos_first = _wenoz3(h[1:-1, 0:1], h[1:-1, 1:2], h[1:-1, 2:3])
    # WENO-Z-3 fallback at last interior column (i=Nx-2)
    h3_pos_last = _wenoz3(h[1:-1, -3:-2], h[1:-1, -2:-1], h[1:-1, -1:])
    h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=1)
    # WENO-Z-5 right-biased: valid for i=1..Nx-4
    h5_neg_int = _wenoz5_right(
        h[1:-1, :-4], h[1:-1, 1:-3], h[1:-1, 2:-2], h[1:-1, 3:-1], h[1:-1, 4:]
    )
    # WENO-Z-3 right-biased fallback at column i=Nx-3
    h3_neg_penultimate = _wenoz3_right(h[1:-1, -3:-2], h[1:-1, -2:-1], h[1:-1, -1:])
    # 1st-order upwind fallback at east boundary column (i=Nx-2)
    h1_neg_last = h[1:-1, -1:]
    h_neg = jnp.concatenate([h5_neg_int, h3_neg_penultimate, h1_neg_last], axis=1)
    h_face = jnp.where(u[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1], h)
    return out

wenoz5_x_masked(h, u, mask)

5-point WENO-Z east-face flux with mask-aware adaptive stencil selection.

Delegates to :func:~finitevolx._src.advection.flux.upwind_flux with a three-tier hierarchy (upwind1 / WENO-Z-3 / WENO-Z-5).

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Cell-centre tracer field.

 required
u Float[Array, 'Ny Nx']

East-face velocity.

 required
mask ArakawaCGridMask

Arakawa C-grid mask providing stencil-capability information.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

East-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
def wenoz5_x_masked(
    self,
    h: Float[Array, "Ny Nx"],
    u: Float[Array, "Ny Nx"],
    mask: ArakawaCGridMask,
) -> Float[Array, "Ny Nx"]:
    """5-point WENO-Z east-face flux with mask-aware adaptive stencil selection.

    Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
    three-tier hierarchy (upwind1 / WENO-Z-3 / WENO-Z-5).

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Cell-centre tracer field.
    u : Float[Array, "Ny Nx"]
        East-face velocity.
    mask : ArakawaCGridMask
        Arakawa C-grid mask providing stencil-capability information.

    Returns
    -------
    Float[Array, "Ny Nx"]
        East-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4, 6))
    rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
        2: self.upwind1_x,
        4: self.wenoz3_x,
        6: self.wenoz5_x,
    }
    return upwind_flux(h, u, dim=1, rec_funcs=rec_funcs, mask_hierarchy=amasks)

wenoz5_y(h, v)

5-point WENO-Z north-face flux with sign-dependent boundary fallbacks.

Positive flow: fn[j+1/2,i] = WENOZ5(h[j-2..j+2,i]) * v for j = 2..Ny-3; WENO-Z-3 fallback at j = 1 and j = Ny-2. Negative flow: fn[j+1/2,i] = WENOZ5_right(h[j-1,i], h[j,i], h[j+1,i], h[j+2,i], h[j+3,i]) * v for j = 1..Ny-4; WENO-Z-3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.

Source code in finitevolx/_src/advection/reconstruction.py 
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
def wenoz5_y(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
) -> Float[Array, "Ny Nx"]:
    """5-point WENO-Z north-face flux with sign-dependent boundary fallbacks.

    Positive flow:  fn[j+1/2,i] = WENOZ5(h[j-2..j+2,i]) * v
        for j = 2..Ny-3; WENO-Z-3 fallback at j = 1 and j = Ny-2.
    Negative flow:  fn[j+1/2,i] = WENOZ5_right(h[j-1,i], h[j,i], h[j+1,i], h[j+2,i], h[j+3,i]) * v
        for j = 1..Ny-4; WENO-Z-3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.
    """
    # WENO-Z-5 positive: valid for j=2..Ny-3
    h5_pos_int = _wenoz5(
        h[:-4, 1:-1], h[1:-3, 1:-1], h[2:-2, 1:-1], h[3:-1, 1:-1], h[4:, 1:-1]
    )
    # WENO-Z-3 fallback at first interior row (j=1)
    h3_pos_first = _wenoz3(h[0:1, 1:-1], h[1:2, 1:-1], h[2:3, 1:-1])
    # WENO-Z-3 fallback at last interior row (j=Ny-2)
    h3_pos_last = _wenoz3(h[-3:-2, 1:-1], h[-2:-1, 1:-1], h[-1:, 1:-1])
    h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=0)
    # WENO-Z-5 right-biased: valid for j=1..Ny-4
    h5_neg_int = _wenoz5_right(
        h[:-4, 1:-1], h[1:-3, 1:-1], h[2:-2, 1:-1], h[3:-1, 1:-1], h[4:, 1:-1]
    )
    # WENO-Z-3 right-biased fallback at row j=Ny-3
    h3_neg_penultimate = _wenoz3_right(h[-3:-2, 1:-1], h[-2:-1, 1:-1], h[-1:, 1:-1])
    # 1st-order upwind fallback at north boundary row (j=Ny-2)
    h1_neg_last = h[-1:, 1:-1]
    h_neg = jnp.concatenate([h5_neg_int, h3_neg_penultimate, h1_neg_last], axis=0)
    h_face = jnp.where(v[1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1], h)
    return out

wenoz5_y_masked(h, v, mask)

5-point WENO-Z north-face flux with mask-aware adaptive stencil selection.

Delegates to :func:~finitevolx._src.advection.flux.upwind_flux with a three-tier hierarchy (upwind1 / WENO-Z-3 / WENO-Z-5).

Parameters:

Name Type Description Default
h Float[Array, 'Ny Nx']

Cell-centre tracer field.

 required
v Float[Array, 'Ny Nx']

North-face velocity.

 required
mask ArakawaCGridMask

Arakawa C-grid mask providing stencil-capability information.

 required

Returns:

Type Description
Float[Array, 'Ny Nx']

North-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
def wenoz5_y_masked(
    self,
    h: Float[Array, "Ny Nx"],
    v: Float[Array, "Ny Nx"],
    mask: ArakawaCGridMask,
) -> Float[Array, "Ny Nx"]:
    """5-point WENO-Z north-face flux with mask-aware adaptive stencil selection.

    Delegates to :func:`~finitevolx._src.advection.flux.upwind_flux` with a
    three-tier hierarchy (upwind1 / WENO-Z-3 / WENO-Z-5).

    Parameters
    ----------
    h : Float[Array, "Ny Nx"]
        Cell-centre tracer field.
    v : Float[Array, "Ny Nx"]
        North-face velocity.
    mask : ArakawaCGridMask
        Arakawa C-grid mask providing stencil-capability information.

    Returns
    -------
    Float[Array, "Ny Nx"]
        North-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4, 6))
    rec_funcs: dict[int, Callable[..., Float[Array, "Ny Nx"]]] = {
        2: self.upwind1_y,
        4: self.wenoz3_y,
        6: self.wenoz5_y,
    }
    return upwind_flux(h, v, dim=0, rec_funcs=rec_funcs, mask_hierarchy=amasks)

finitevolx.Reconstruction3D

Bases: Module

3-D face-value reconstruction.

Parameters:

Name Type Description Default
grid ArakawaCGrid3D
required
Source code in finitevolx/_src/advection/reconstruction.py 
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
class Reconstruction3D(eqx.Module):
    """3-D face-value reconstruction.

    Parameters
    ----------
    grid : ArakawaCGrid3D
    """

    grid: ArakawaCGrid3D

    def naive_x(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """Naive east-face flux over all z-levels.

        fe[k, j, i+1/2] = 1/2 * (h[k,j,i] + h[k,j,i+1]) * u[k,j,i+1/2]
        """
        out = interior(
            0.5 * (h[1:-1, 1:-1, 1:-1] + h[1:-1, 1:-1, 2:]) * u[1:-1, 1:-1, 1:-1],
            h,
        )
        return out

    def naive_y(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """Naive north-face flux over all z-levels.

        fn[k, j+1/2, i] = 1/2 * (h[k,j,i] + h[k,j+1,i]) * v[k,j+1/2,i]
        """
        out = interior(
            0.5 * (h[1:-1, 1:-1, 1:-1] + h[1:-1, 2:, 1:-1]) * v[1:-1, 1:-1, 1:-1],
            h,
        )
        return out

    def upwind1_x(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """1st-order upwind east-face flux over all z-levels."""
        h_face = jnp.where(
            u[1:-1, 1:-1, 1:-1] >= 0.0,
            h[1:-1, 1:-1, 1:-1],
            h[1:-1, 1:-1, 2:],
        )
        out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
        return out

    def upwind1_y(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """1st-order upwind north-face flux over all z-levels."""
        h_face = jnp.where(
            v[1:-1, 1:-1, 1:-1] >= 0.0,
            h[1:-1, 1:-1, 1:-1],
            h[1:-1, 2:, 1:-1],
        )
        out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
        return out

    def weno3_x(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """3-point WENO east-face flux over all z-levels with boundary fallback.

        Positive flow:  fe[k,j,i+1/2] = WENO3(h[k,j,i-1], h[k,j,i],   h[k,j,i+1]) * u
        Negative flow:  fe[k,j,i+1/2] = WENO3_right(h[k,j,i], h[k,j,i+1], h[k,j,i+2]) * u
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.
        """
        # WENO-3 positive: left-biased, valid for all interior faces
        h_pos = _weno3(h[1:-1, 1:-1, :-2], h[1:-1, 1:-1, 1:-1], h[1:-1, 1:-1, 2:])
        # WENO-3 negative: right-biased, valid for i+2 < Nx
        h_neg_interior = _weno3_right(
            h[1:-1, 1:-1, 1:-2], h[1:-1, 1:-1, 2:-1], h[1:-1, 1:-1, 3:]
        )
        # 1st-order upwind fallback at east boundary
        h_neg_boundary = h[1:-1, 1:-1, -1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=2)
        h_face = jnp.where(u[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
        return out

    def weno3_y(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """3-point WENO north-face flux over all z-levels with boundary fallback.

        Positive flow:  fn[k,j+1/2,i] = WENO3(h[k,j-1,i], h[k,j,i],   h[k,j+1,i]) * v
        Negative flow:  fn[k,j+1/2,i] = WENO3_right(h[k,j,i], h[k,j+1,i], h[k,j+2,i]) * v
        Falls back to 1st-order upwind on north boundary where j+2 unavailable.
        """
        # WENO-3 positive: left-biased, valid for all interior faces
        h_pos = _weno3(h[1:-1, :-2, 1:-1], h[1:-1, 1:-1, 1:-1], h[1:-1, 2:, 1:-1])
        # WENO-3 negative: right-biased, valid for j+2 < Ny
        h_neg_interior = _weno3_right(
            h[1:-1, 1:-2, 1:-1], h[1:-1, 2:-1, 1:-1], h[1:-1, 3:, 1:-1]
        )
        # 1st-order upwind fallback at north boundary
        h_neg_boundary = h[1:-1, -1:, 1:-1]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
        h_face = jnp.where(v[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
        return out

    def wenoz3_x(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """3-point WENO-Z east-face flux over all z-levels with boundary fallback.

        Positive flow:  fe[k,j,i+1/2] = WENOZ3(h[k,j,i-1], h[k,j,i],   h[k,j,i+1]) * u
        Negative flow:  fe[k,j,i+1/2] = WENOZ3_right(h[k,j,i], h[k,j,i+1], h[k,j,i+2]) * u
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.
        """
        # WENO-Z-3 positive: left-biased, valid for all interior faces
        h_pos = _wenoz3(h[1:-1, 1:-1, :-2], h[1:-1, 1:-1, 1:-1], h[1:-1, 1:-1, 2:])
        # WENO-Z-3 negative: right-biased, valid for i+2 < Nx
        h_neg_interior = _wenoz3_right(
            h[1:-1, 1:-1, 1:-2], h[1:-1, 1:-1, 2:-1], h[1:-1, 1:-1, 3:]
        )
        # 1st-order upwind fallback at east boundary
        h_neg_boundary = h[1:-1, 1:-1, -1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=2)
        h_face = jnp.where(u[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
        return out

    def wenoz3_y(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """3-point WENO-Z north-face flux over all z-levels with boundary fallback.

        Positive flow:  fn[k,j+1/2,i] = WENOZ3(h[k,j-1,i], h[k,j,i],   h[k,j+1,i]) * v
        Negative flow:  fn[k,j+1/2,i] = WENOZ3_right(h[k,j,i], h[k,j+1,i], h[k,j+2,i]) * v
        Falls back to 1st-order upwind on north boundary where j+2 unavailable.
        """
        # WENO-Z-3 positive: left-biased, valid for all interior faces
        h_pos = _wenoz3(h[1:-1, :-2, 1:-1], h[1:-1, 1:-1, 1:-1], h[1:-1, 2:, 1:-1])
        # WENO-Z-3 negative: right-biased, valid for j+2 < Ny
        h_neg_interior = _wenoz3_right(
            h[1:-1, 1:-2, 1:-1], h[1:-1, 2:-1, 1:-1], h[1:-1, 3:, 1:-1]
        )
        # 1st-order upwind fallback at north boundary
        h_neg_boundary = h[1:-1, -1:, 1:-1]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
        h_face = jnp.where(v[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
        return out

    def weno5_x(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """5-point WENO east-face flux over all z-levels with sign-dependent fallbacks.

        Positive flow:  fe[k,j,i+1/2] = WENO5(h[k,j,i-2..i+2]) * u
            for i = 2..Nx-3; WENO3 fallback at i = 1 and i = Nx-2.
        Negative flow:  fe[k,j,i+1/2] = WENO5_right(h[k,j,i-1], h[k,j,i], h[k,j,i+1], h[k,j,i+2], h[k,j,i+3]) * u
            for i = 1..Nx-4; WENO3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
        """
        h5_pos_int = _weno5(
            h[1:-1, 1:-1, :-4],
            h[1:-1, 1:-1, 1:-3],
            h[1:-1, 1:-1, 2:-2],
            h[1:-1, 1:-1, 3:-1],
            h[1:-1, 1:-1, 4:],
        )
        h3_pos_first = _weno3(
            h[1:-1, 1:-1, 0:1], h[1:-1, 1:-1, 1:2], h[1:-1, 1:-1, 2:3]
        )
        h3_pos_last = _weno3(
            h[1:-1, 1:-1, -3:-2], h[1:-1, 1:-1, -2:-1], h[1:-1, 1:-1, -1:]
        )
        h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=2)
        h5_neg_int = _weno5_right(
            h[1:-1, 1:-1, :-4],
            h[1:-1, 1:-1, 1:-3],
            h[1:-1, 1:-1, 2:-2],
            h[1:-1, 1:-1, 3:-1],
            h[1:-1, 1:-1, 4:],
        )
        h3_neg_penultimate = _weno3_right(
            h[1:-1, 1:-1, -3:-2], h[1:-1, 1:-1, -2:-1], h[1:-1, 1:-1, -1:]
        )
        h_neg = jnp.concatenate(
            [h5_neg_int, h3_neg_penultimate, h[1:-1, 1:-1, -1:]], axis=2
        )
        h_face = jnp.where(u[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
        return out

    def weno5_y(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """5-point WENO north-face flux over all z-levels with sign-dependent fallbacks.

        Positive flow:  fn[k,j+1/2,i] = WENO5(h[k,j-2..j+2,i]) * v
            for j = 2..Ny-3; WENO3 fallback at j = 1 and j = Ny-2.
        Negative flow:  fn[k,j+1/2,i] = WENO5_right(h[k,j-1,i], h[k,j,i], h[k,j+1,i], h[k,j+2,i], h[k,j+3,i]) * v
            for j = 1..Ny-4; WENO3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.
        """
        h5_pos_int = _weno5(
            h[1:-1, :-4, 1:-1],
            h[1:-1, 1:-3, 1:-1],
            h[1:-1, 2:-2, 1:-1],
            h[1:-1, 3:-1, 1:-1],
            h[1:-1, 4:, 1:-1],
        )
        h3_pos_first = _weno3(
            h[1:-1, 0:1, 1:-1], h[1:-1, 1:2, 1:-1], h[1:-1, 2:3, 1:-1]
        )
        h3_pos_last = _weno3(
            h[1:-1, -3:-2, 1:-1], h[1:-1, -2:-1, 1:-1], h[1:-1, -1:, 1:-1]
        )
        h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=1)
        h5_neg_int = _weno5_right(
            h[1:-1, :-4, 1:-1],
            h[1:-1, 1:-3, 1:-1],
            h[1:-1, 2:-2, 1:-1],
            h[1:-1, 3:-1, 1:-1],
            h[1:-1, 4:, 1:-1],
        )
        h3_neg_penultimate = _weno3_right(
            h[1:-1, -3:-2, 1:-1], h[1:-1, -2:-1, 1:-1], h[1:-1, -1:, 1:-1]
        )
        h_neg = jnp.concatenate(
            [h5_neg_int, h3_neg_penultimate, h[1:-1, -1:, 1:-1]], axis=1
        )
        h_face = jnp.where(v[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
        return out

    def wenoz5_x(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """5-point WENO-Z east-face flux over all z-levels with sign-dependent fallbacks.

        Positive flow:  fe[k,j,i+1/2] = WENOZ5(h[k,j,i-2..i+2]) * u
            for i = 2..Nx-3; WENO-Z-3 fallback at i = 1 and i = Nx-2.
        Negative flow:  fe[k,j,i+1/2] = WENOZ5_right(h[k,j,i-1], h[k,j,i], h[k,j,i+1], h[k,j,i+2], h[k,j,i+3]) * u
            for i = 1..Nx-4; WENO-Z-3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
        """
        h5_pos_int = _wenoz5(
            h[1:-1, 1:-1, :-4],
            h[1:-1, 1:-1, 1:-3],
            h[1:-1, 1:-1, 2:-2],
            h[1:-1, 1:-1, 3:-1],
            h[1:-1, 1:-1, 4:],
        )
        h3_pos_first = _wenoz3(
            h[1:-1, 1:-1, 0:1], h[1:-1, 1:-1, 1:2], h[1:-1, 1:-1, 2:3]
        )
        h3_pos_last = _wenoz3(
            h[1:-1, 1:-1, -3:-2], h[1:-1, 1:-1, -2:-1], h[1:-1, 1:-1, -1:]
        )
        h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=2)
        h5_neg_int = _wenoz5_right(
            h[1:-1, 1:-1, :-4],
            h[1:-1, 1:-1, 1:-3],
            h[1:-1, 1:-1, 2:-2],
            h[1:-1, 1:-1, 3:-1],
            h[1:-1, 1:-1, 4:],
        )
        h3_neg_penultimate = _wenoz3_right(
            h[1:-1, 1:-1, -3:-2], h[1:-1, 1:-1, -2:-1], h[1:-1, 1:-1, -1:]
        )
        h_neg = jnp.concatenate(
            [h5_neg_int, h3_neg_penultimate, h[1:-1, 1:-1, -1:]], axis=2
        )
        h_face = jnp.where(u[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
        return out

    def wenoz5_y(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """5-point WENO-Z north-face flux over all z-levels with sign-dependent fallbacks.

        Positive flow:  fn[k,j+1/2,i] = WENOZ5(h[k,j-2..j+2,i]) * v
            for j = 2..Ny-3; WENO-Z-3 fallback at j = 1 and j = Ny-2.
        Negative flow:  fn[k,j+1/2,i] = WENOZ5_right(h[k,j-1,i], h[k,j,i], h[k,j+1,i], h[k,j+2,i], h[k,j+3,i]) * v
            for j = 1..Ny-4; WENO-Z-3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.
        """
        h5_pos_int = _wenoz5(
            h[1:-1, :-4, 1:-1],
            h[1:-1, 1:-3, 1:-1],
            h[1:-1, 2:-2, 1:-1],
            h[1:-1, 3:-1, 1:-1],
            h[1:-1, 4:, 1:-1],
        )
        h3_pos_first = _wenoz3(
            h[1:-1, 0:1, 1:-1], h[1:-1, 1:2, 1:-1], h[1:-1, 2:3, 1:-1]
        )
        h3_pos_last = _wenoz3(
            h[1:-1, -3:-2, 1:-1], h[1:-1, -2:-1, 1:-1], h[1:-1, -1:, 1:-1]
        )
        h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=1)
        h5_neg_int = _wenoz5_right(
            h[1:-1, :-4, 1:-1],
            h[1:-1, 1:-3, 1:-1],
            h[1:-1, 2:-2, 1:-1],
            h[1:-1, 3:-1, 1:-1],
            h[1:-1, 4:, 1:-1],
        )
        h3_neg_penultimate = _wenoz3_right(
            h[1:-1, -3:-2, 1:-1], h[1:-1, -2:-1, 1:-1], h[1:-1, -1:, 1:-1]
        )
        h_neg = jnp.concatenate(
            [h5_neg_int, h3_neg_penultimate, h[1:-1, -1:, 1:-1]], axis=1
        )
        h_face = jnp.where(v[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
        return out

    def weno7_x(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """7-point WENO east-face flux over all z-levels with hierarchical fallbacks."""
        return _weno_flux_axis_3d_x(h, u, order=7)

    def weno7_y(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """7-point WENO north-face flux over all z-levels with hierarchical fallbacks."""
        return _weno_flux_axis_3d_y(h, v, order=7)

    def weno9_x(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """9-point WENO east-face flux over all z-levels with hierarchical fallbacks."""
        return _weno_flux_axis_3d_x(h, u, order=9)

    def weno9_y(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
    ) -> Float[Array, "Nz Ny Nx"]:
        """9-point WENO north-face flux over all z-levels with hierarchical fallbacks."""
        return _weno_flux_axis_3d_y(h, v, order=9)

    def tvd_x(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
        limiter: str = "minmod",
    ) -> Float[Array, "Nz Ny Nx"]:
        """TVD east-face flux over all z-levels using a flux limiter.

        Positive flow:
            r           = (h[k,j,i] − h[k,j,i−1]) / (h[k,j,i+1] − h[k,j,i])
            h_L[k,j,i+½] = h[k,j,i]   + ½ φ(r) (h[k,j,i+1] − h[k,j,i])

        Negative flow:
            r           = (h[k,j,i+1] − h[k,j,i+2]) / (h[k,j,i] − h[k,j,i+1])
            h_R[k,j,i+½] = h[k,j,i+1] + ½ φ(r) (h[k,j,i]   − h[k,j,i+1])
        Falls back to 1st-order upwind on east boundary where i+2 unavailable.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Scalar at T-points.
        u : Float[Array, "Nz Ny Nx"]
            x-velocity at U-points.
        limiter : str
            Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
            or ``'mc'``.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            East-face flux with zero ghost ring.
        """
        phi = _get_limiter(limiter)
        # Positive flow: h_face = h[k,j,i] + 0.5*phi(r)*(h[k,j,i+1]-h[k,j,i])
        diff_fwd = h[1:-1, 1:-1, 2:] - h[1:-1, 1:-1, 1:-1]
        diff_bwd = h[1:-1, 1:-1, 1:-1] - h[1:-1, 1:-1, :-2]
        r_pos = diff_bwd / (diff_fwd + _TVD_EPS)
        h_pos = h[1:-1, 1:-1, 1:-1] + 0.5 * phi(r_pos) * diff_fwd
        # Negative flow: h_face = h[k,j,i+1] + 0.5*phi(r)*(h[k,j,i]-h[k,j,i+1])
        diff_neg = h[1:-1, 1:-1, 1:-2] - h[1:-1, 1:-1, 2:-1]
        diff_neg2 = h[1:-1, 1:-1, 2:-1] - h[1:-1, 1:-1, 3:]
        r_neg_int = diff_neg2 / (diff_neg + _TVD_EPS)
        h_neg_interior = h[1:-1, 1:-1, 2:-1] + 0.5 * phi(r_neg_int) * diff_neg
        # 1st-order upwind fallback at east boundary column (i=Nx-2)
        h_neg_boundary = h[1:-1, 1:-1, -1:]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=2)
        h_face = jnp.where(u[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
        return out

    def tvd_y(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
        limiter: str = "minmod",
    ) -> Float[Array, "Nz Ny Nx"]:
        """TVD north-face flux over all z-levels using a flux limiter.

        Positive flow:
            r           = (h[k,j,i] − h[k,j−1,i]) / (h[k,j+1,i] − h[k,j,i])
            h_L[k,j+½,i] = h[k,j,i]   + ½ φ(r) (h[k,j+1,i] − h[k,j,i])

        Negative flow:
            r           = (h[k,j+1,i] − h[k,j+2,i]) / (h[k,j,i] − h[k,j+1,i])
            h_R[k,j+½,i] = h[k,j+1,i] + ½ φ(r) (h[k,j,i]   − h[k,j+1,i])
        Falls back to 1st-order upwind on north boundary where j+2 unavailable.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Scalar at T-points.
        v : Float[Array, "Nz Ny Nx"]
            y-velocity at V-points.
        limiter : str
            Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
            or ``'mc'``.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            North-face flux with zero ghost ring.
        """
        phi = _get_limiter(limiter)
        # Positive flow: h_face = h[k,j,i] + 0.5*phi(r)*(h[k,j+1,i]-h[k,j,i])
        diff_fwd = h[1:-1, 2:, 1:-1] - h[1:-1, 1:-1, 1:-1]
        diff_bwd = h[1:-1, 1:-1, 1:-1] - h[1:-1, :-2, 1:-1]
        r_pos = diff_bwd / (diff_fwd + _TVD_EPS)
        h_pos = h[1:-1, 1:-1, 1:-1] + 0.5 * phi(r_pos) * diff_fwd
        # Negative flow: h_face = h[k,j+1,i] + 0.5*phi(r)*(h[k,j,i]-h[k,j+1,i])
        diff_neg = h[1:-1, 1:-2, 1:-1] - h[1:-1, 2:-1, 1:-1]
        diff_neg2 = h[1:-1, 2:-1, 1:-1] - h[1:-1, 3:, 1:-1]
        r_neg_int = diff_neg2 / (diff_neg + _TVD_EPS)
        h_neg_interior = h[1:-1, 2:-1, 1:-1] + 0.5 * phi(r_neg_int) * diff_neg
        # 1st-order upwind fallback at north boundary row (j=Ny-2)
        h_neg_boundary = h[1:-1, -1:, 1:-1]
        h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
        h_face = jnp.where(v[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
        out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
        return out

    def tvd_x_masked(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
        mask: ArakawaCGridMask,
        limiter: str = "minmod",
    ) -> Float[Array, "Nz Ny Nx"]:
        """TVD east-face flux over all z-levels with mask-aware stencil selection.

        The 2-D ``mask`` is broadcast over the z-dimension.  Falls back to
        1st-order upwind near coastlines where the TVD stencil crosses land.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Cell-centre tracer field.
        u : Float[Array, "Nz Ny Nx"]
            East-face velocity.
        mask : ArakawaCGridMask
            2-D Arakawa C-grid mask (broadcast over z).
        limiter : str
            Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
            or ``'mc'``.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            East-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4))
        m_tvd = amasks[4]
        fe_tvd = self.tvd_x(h, u, limiter=limiter)
        fe_u1 = self.upwind1_x(h, u)
        pos_flow = u[1:-1, 1:-1, 1:-1] >= 0.0
        use_tvd = jnp.where(pos_flow, m_tvd[None, 1:-1, 1:-1], m_tvd[None, 1:-1, 2:])
        selected = jnp.where(use_tvd, fe_tvd[1:-1, 1:-1, 1:-1], fe_u1[1:-1, 1:-1, 1:-1])
        out = interior(selected, h)
        return out

    def tvd_y_masked(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
        mask: ArakawaCGridMask,
        limiter: str = "minmod",
    ) -> Float[Array, "Nz Ny Nx"]:
        """TVD north-face flux over all z-levels with mask-aware stencil selection.

        The 2-D ``mask`` is broadcast over the z-dimension.  Falls back to
        1st-order upwind near coastlines where the TVD stencil crosses land.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Cell-centre tracer field.
        v : Float[Array, "Nz Ny Nx"]
            North-face velocity.
        mask : ArakawaCGridMask
            2-D Arakawa C-grid mask (broadcast over z).
        limiter : str
            Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
            or ``'mc'``.

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            North-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4))
        m_tvd = amasks[4]
        fn_tvd = self.tvd_y(h, v, limiter=limiter)
        fn_u1 = self.upwind1_y(h, v)
        pos_flow = v[1:-1, 1:-1, 1:-1] >= 0.0
        use_tvd = jnp.where(pos_flow, m_tvd[None, 1:-1, 1:-1], m_tvd[None, 2:, 1:-1])
        selected = jnp.where(use_tvd, fn_tvd[1:-1, 1:-1, 1:-1], fn_u1[1:-1, 1:-1, 1:-1])
        out = interior(selected, h)
        return out

    def weno3_x_masked(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
        mask: ArakawaCGridMask,
    ) -> Float[Array, "Nz Ny Nx"]:
        """3-point WENO east-face flux over all z-levels with mask-aware stencil selection.

        The 2-D ``mask`` is broadcast over the z-dimension.  Falls back to
        1st-order upwind near coastlines where the WENO3 stencil crosses land.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Cell-centre tracer field.
        u : Float[Array, "Nz Ny Nx"]
            East-face velocity.
        mask : ArakawaCGridMask
            2-D Arakawa C-grid mask (broadcast over z).

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            East-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4))
        m_w3 = amasks[4]
        fe_w3 = self.weno3_x(h, u)
        fe_u1 = self.upwind1_x(h, u)
        pos_flow = u[1:-1, 1:-1, 1:-1] >= 0.0
        use_w3 = jnp.where(pos_flow, m_w3[None, 1:-1, 1:-1], m_w3[None, 1:-1, 2:])
        selected = jnp.where(use_w3, fe_w3[1:-1, 1:-1, 1:-1], fe_u1[1:-1, 1:-1, 1:-1])
        out = interior(selected, h)
        return out

    def weno3_y_masked(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
        mask: ArakawaCGridMask,
    ) -> Float[Array, "Nz Ny Nx"]:
        """3-point WENO north-face flux over all z-levels with mask-aware stencil selection.

        The 2-D ``mask`` is broadcast over the z-dimension.  Falls back to
        1st-order upwind near coastlines where the WENO3 stencil crosses land.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Cell-centre tracer field.
        v : Float[Array, "Nz Ny Nx"]
            North-face velocity.
        mask : ArakawaCGridMask
            2-D Arakawa C-grid mask (broadcast over z).

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            North-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4))
        m_w3 = amasks[4]
        fn_w3 = self.weno3_y(h, v)
        fn_u1 = self.upwind1_y(h, v)
        pos_flow = v[1:-1, 1:-1, 1:-1] >= 0.0
        use_w3 = jnp.where(pos_flow, m_w3[None, 1:-1, 1:-1], m_w3[None, 2:, 1:-1])
        selected = jnp.where(use_w3, fn_w3[1:-1, 1:-1, 1:-1], fn_u1[1:-1, 1:-1, 1:-1])
        out = interior(selected, h)
        return out

    def weno5_x_masked(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
        mask: ArakawaCGridMask,
    ) -> Float[Array, "Nz Ny Nx"]:
        """5-point WENO east-face flux over all z-levels with mask-aware adaptive stencil.

        The 2-D ``mask`` is broadcast over the z-dimension so that the same
        horizontal stencil-capability map is applied at every depth level.
        Adaptively falls back to WENO3 or 1st-order upwind near coastlines.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Cell-centre tracer field.
        u : Float[Array, "Nz Ny Nx"]
            East-face velocity.
        mask : ArakawaCGridMask
            2-D Arakawa C-grid mask (broadcast over z).

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            East-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4, 6))
        m5 = amasks[6]  # (Ny, Nx)
        m3 = amasks[4]
        fe_w5 = self.weno5_x(h, u)
        fe_w3 = self.weno3_x(h, u)
        fe_u1 = self.upwind1_x(h, u)
        # Select tier based on upwind-cell stencil capability, broadcast mask over z
        # Positive flow: upwind cell is (j, i)   → mask at [1:-1, 1:-1]
        # Negative flow: upwind cell is (j, i+1) → mask at [1:-1, 2:]
        pos_flow = u[1:-1, 1:-1, 1:-1] >= 0.0
        use_w5 = jnp.where(pos_flow, m5[None, 1:-1, 1:-1], m5[None, 1:-1, 2:])
        use_w3 = jnp.where(pos_flow, m3[None, 1:-1, 1:-1], m3[None, 1:-1, 2:])
        selected = jnp.where(
            use_w5,
            fe_w5[1:-1, 1:-1, 1:-1],
            jnp.where(use_w3, fe_w3[1:-1, 1:-1, 1:-1], fe_u1[1:-1, 1:-1, 1:-1]),
        )
        out = interior(selected, h)
        return out

    def weno5_y_masked(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
        mask: ArakawaCGridMask,
    ) -> Float[Array, "Nz Ny Nx"]:
        """5-point WENO north-face flux over all z-levels with mask-aware adaptive stencil.

        The 2-D ``mask`` is broadcast over the z-dimension so that the same
        horizontal stencil-capability map is applied at every depth level.
        Adaptively falls back to WENO3 or 1st-order upwind near coastlines.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Cell-centre tracer field.
        v : Float[Array, "Nz Ny Nx"]
            North-face velocity.
        mask : ArakawaCGridMask
            2-D Arakawa C-grid mask (broadcast over z).

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            North-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4, 6))
        m5 = amasks[6]
        m3 = amasks[4]
        fn_w5 = self.weno5_y(h, v)
        fn_w3 = self.weno3_y(h, v)
        fn_u1 = self.upwind1_y(h, v)
        # Select tier based on upwind-cell stencil capability, broadcast mask over z
        # Positive flow: upwind cell is (j, i)   → mask at [1:-1, 1:-1]
        # Negative flow: upwind cell is (j+1, i) → mask at [2:, 1:-1]
        pos_flow = v[1:-1, 1:-1, 1:-1] >= 0.0
        use_w5 = jnp.where(pos_flow, m5[None, 1:-1, 1:-1], m5[None, 2:, 1:-1])
        use_w3 = jnp.where(pos_flow, m3[None, 1:-1, 1:-1], m3[None, 2:, 1:-1])
        selected = jnp.where(
            use_w5,
            fn_w5[1:-1, 1:-1, 1:-1],
            jnp.where(use_w3, fn_w3[1:-1, 1:-1, 1:-1], fn_u1[1:-1, 1:-1, 1:-1]),
        )
        out = interior(selected, h)
        return out

    def wenoz5_x_masked(
        self,
        h: Float[Array, "Nz Ny Nx"],
        u: Float[Array, "Nz Ny Nx"],
        mask: ArakawaCGridMask,
    ) -> Float[Array, "Nz Ny Nx"]:
        """5-point WENO-Z east-face flux over all z-levels with mask-aware adaptive stencil.

        The 2-D ``mask`` is broadcast over the z-dimension so that the same
        horizontal stencil-capability map is applied at every depth level.
        Adaptively falls back to WENO-Z-3 or 1st-order upwind near coastlines.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Cell-centre tracer field.
        u : Float[Array, "Nz Ny Nx"]
            East-face velocity.
        mask : ArakawaCGridMask
            2-D Arakawa C-grid mask (broadcast over z).

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            East-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4, 6))
        m5 = amasks[6]
        m3 = amasks[4]
        fe_w5 = self.wenoz5_x(h, u)
        fe_w3 = self.wenoz3_x(h, u)
        fe_u1 = self.upwind1_x(h, u)
        pos_flow = u[1:-1, 1:-1, 1:-1] >= 0.0
        use_w5 = jnp.where(pos_flow, m5[None, 1:-1, 1:-1], m5[None, 1:-1, 2:])
        use_w3 = jnp.where(pos_flow, m3[None, 1:-1, 1:-1], m3[None, 1:-1, 2:])
        selected = jnp.where(
            use_w5,
            fe_w5[1:-1, 1:-1, 1:-1],
            jnp.where(use_w3, fe_w3[1:-1, 1:-1, 1:-1], fe_u1[1:-1, 1:-1, 1:-1]),
        )
        out = interior(selected, h)
        return out

    def wenoz5_y_masked(
        self,
        h: Float[Array, "Nz Ny Nx"],
        v: Float[Array, "Nz Ny Nx"],
        mask: ArakawaCGridMask,
    ) -> Float[Array, "Nz Ny Nx"]:
        """5-point WENO-Z north-face flux over all z-levels with mask-aware adaptive stencil.

        The 2-D ``mask`` is broadcast over the z-dimension so that the same
        horizontal stencil-capability map is applied at every depth level.
        Adaptively falls back to WENO-Z-3 or 1st-order upwind near coastlines.

        Parameters
        ----------
        h : Float[Array, "Nz Ny Nx"]
            Cell-centre tracer field.
        v : Float[Array, "Nz Ny Nx"]
            North-face velocity.
        mask : ArakawaCGridMask
            2-D Arakawa C-grid mask (broadcast over z).

        Returns
        -------
        Float[Array, "Nz Ny Nx"]
            North-face flux with zero ghost ring.
        """
        amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4, 6))
        m5 = amasks[6]
        m3 = amasks[4]
        fn_w5 = self.wenoz5_y(h, v)
        fn_w3 = self.wenoz3_y(h, v)
        fn_u1 = self.upwind1_y(h, v)
        pos_flow = v[1:-1, 1:-1, 1:-1] >= 0.0
        use_w5 = jnp.where(pos_flow, m5[None, 1:-1, 1:-1], m5[None, 2:, 1:-1])
        use_w3 = jnp.where(pos_flow, m3[None, 1:-1, 1:-1], m3[None, 2:, 1:-1])
        selected = jnp.where(
            use_w5,
            fn_w5[1:-1, 1:-1, 1:-1],
            jnp.where(use_w3, fn_w3[1:-1, 1:-1, 1:-1], fn_u1[1:-1, 1:-1, 1:-1]),
        )
        out = interior(selected, h)
        return out

naive_x(h, u)

Naive east-face flux over all z-levels.

fe[k, j, i+1/2] = 1/2 * (h[k,j,i] + h[k,j,i+1]) * u[k,j,i+1/2]

Source code in finitevolx/_src/advection/reconstruction.py 
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
def naive_x(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """Naive east-face flux over all z-levels.

    fe[k, j, i+1/2] = 1/2 * (h[k,j,i] + h[k,j,i+1]) * u[k,j,i+1/2]
    """
    out = interior(
        0.5 * (h[1:-1, 1:-1, 1:-1] + h[1:-1, 1:-1, 2:]) * u[1:-1, 1:-1, 1:-1],
        h,
    )
    return out

naive_y(h, v)

Naive north-face flux over all z-levels.

fn[k, j+1/2, i] = 1/2 * (h[k,j,i] + h[k,j+1,i]) * v[k,j+1/2,i]

Source code in finitevolx/_src/advection/reconstruction.py 
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
def naive_y(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """Naive north-face flux over all z-levels.

    fn[k, j+1/2, i] = 1/2 * (h[k,j,i] + h[k,j+1,i]) * v[k,j+1/2,i]
    """
    out = interior(
        0.5 * (h[1:-1, 1:-1, 1:-1] + h[1:-1, 2:, 1:-1]) * v[1:-1, 1:-1, 1:-1],
        h,
    )
    return out

tvd_x(h, u, limiter='minmod')

TVD east-face flux over all z-levels using a flux limiter.

Positive flow: r = (h[k,j,i] − h[k,j,i−1]) / (h[k,j,i+1] − h[k,j,i]) h_L[k,j,i+½] = h[k,j,i] + ½ φ(r) (h[k,j,i+1] − h[k,j,i])

Negative flow: r = (h[k,j,i+1] − h[k,j,i+2]) / (h[k,j,i] − h[k,j,i+1]) h_R[k,j,i+½] = h[k,j,i+1] + ½ φ(r) (h[k,j,i] − h[k,j,i+1]) Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Scalar at T-points.

 required
u Float[Array, 'Nz Ny Nx']

x-velocity at U-points.

 required
limiter str

Flux limiter name: 'minmod', 'van_leer', 'superbee', or 'mc'.

 'minmod'

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

East-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
def tvd_x(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
    limiter: str = "minmod",
) -> Float[Array, "Nz Ny Nx"]:
    """TVD east-face flux over all z-levels using a flux limiter.

    Positive flow:
        r           = (h[k,j,i] − h[k,j,i−1]) / (h[k,j,i+1] − h[k,j,i])
        h_L[k,j,i+½] = h[k,j,i]   + ½ φ(r) (h[k,j,i+1] − h[k,j,i])

    Negative flow:
        r           = (h[k,j,i+1] − h[k,j,i+2]) / (h[k,j,i] − h[k,j,i+1])
        h_R[k,j,i+½] = h[k,j,i+1] + ½ φ(r) (h[k,j,i]   − h[k,j,i+1])
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Scalar at T-points.
    u : Float[Array, "Nz Ny Nx"]
        x-velocity at U-points.
    limiter : str
        Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
        or ``'mc'``.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        East-face flux with zero ghost ring.
    """
    phi = _get_limiter(limiter)
    # Positive flow: h_face = h[k,j,i] + 0.5*phi(r)*(h[k,j,i+1]-h[k,j,i])
    diff_fwd = h[1:-1, 1:-1, 2:] - h[1:-1, 1:-1, 1:-1]
    diff_bwd = h[1:-1, 1:-1, 1:-1] - h[1:-1, 1:-1, :-2]
    r_pos = diff_bwd / (diff_fwd + _TVD_EPS)
    h_pos = h[1:-1, 1:-1, 1:-1] + 0.5 * phi(r_pos) * diff_fwd
    # Negative flow: h_face = h[k,j,i+1] + 0.5*phi(r)*(h[k,j,i]-h[k,j,i+1])
    diff_neg = h[1:-1, 1:-1, 1:-2] - h[1:-1, 1:-1, 2:-1]
    diff_neg2 = h[1:-1, 1:-1, 2:-1] - h[1:-1, 1:-1, 3:]
    r_neg_int = diff_neg2 / (diff_neg + _TVD_EPS)
    h_neg_interior = h[1:-1, 1:-1, 2:-1] + 0.5 * phi(r_neg_int) * diff_neg
    # 1st-order upwind fallback at east boundary column (i=Nx-2)
    h_neg_boundary = h[1:-1, 1:-1, -1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=2)
    h_face = jnp.where(u[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
    return out

tvd_x_masked(h, u, mask, limiter='minmod')

TVD east-face flux over all z-levels with mask-aware stencil selection.

The 2-D mask is broadcast over the z-dimension. Falls back to 1st-order upwind near coastlines where the TVD stencil crosses land.

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Cell-centre tracer field.

 required
u Float[Array, 'Nz Ny Nx']

East-face velocity.

 required
mask ArakawaCGridMask

2-D Arakawa C-grid mask (broadcast over z).

 required
limiter str

Flux limiter name: 'minmod', 'van_leer', 'superbee', or 'mc'.

 'minmod'

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

East-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
def tvd_x_masked(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
    mask: ArakawaCGridMask,
    limiter: str = "minmod",
) -> Float[Array, "Nz Ny Nx"]:
    """TVD east-face flux over all z-levels with mask-aware stencil selection.

    The 2-D ``mask`` is broadcast over the z-dimension.  Falls back to
    1st-order upwind near coastlines where the TVD stencil crosses land.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Cell-centre tracer field.
    u : Float[Array, "Nz Ny Nx"]
        East-face velocity.
    mask : ArakawaCGridMask
        2-D Arakawa C-grid mask (broadcast over z).
    limiter : str
        Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
        or ``'mc'``.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        East-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4))
    m_tvd = amasks[4]
    fe_tvd = self.tvd_x(h, u, limiter=limiter)
    fe_u1 = self.upwind1_x(h, u)
    pos_flow = u[1:-1, 1:-1, 1:-1] >= 0.0
    use_tvd = jnp.where(pos_flow, m_tvd[None, 1:-1, 1:-1], m_tvd[None, 1:-1, 2:])
    selected = jnp.where(use_tvd, fe_tvd[1:-1, 1:-1, 1:-1], fe_u1[1:-1, 1:-1, 1:-1])
    out = interior(selected, h)
    return out

tvd_y(h, v, limiter='minmod')

TVD north-face flux over all z-levels using a flux limiter.

Positive flow: r = (h[k,j,i] − h[k,j−1,i]) / (h[k,j+1,i] − h[k,j,i]) h_L[k,j+½,i] = h[k,j,i] + ½ φ(r) (h[k,j+1,i] − h[k,j,i])

Negative flow: r = (h[k,j+1,i] − h[k,j+2,i]) / (h[k,j,i] − h[k,j+1,i]) h_R[k,j+½,i] = h[k,j+1,i] + ½ φ(r) (h[k,j,i] − h[k,j+1,i]) Falls back to 1st-order upwind on north boundary where j+2 unavailable.

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Scalar at T-points.

 required
v Float[Array, 'Nz Ny Nx']

y-velocity at V-points.

 required
limiter str

Flux limiter name: 'minmod', 'van_leer', 'superbee', or 'mc'.

 'minmod'

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

North-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
def tvd_y(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
    limiter: str = "minmod",
) -> Float[Array, "Nz Ny Nx"]:
    """TVD north-face flux over all z-levels using a flux limiter.

    Positive flow:
        r           = (h[k,j,i] − h[k,j−1,i]) / (h[k,j+1,i] − h[k,j,i])
        h_L[k,j+½,i] = h[k,j,i]   + ½ φ(r) (h[k,j+1,i] − h[k,j,i])

    Negative flow:
        r           = (h[k,j+1,i] − h[k,j+2,i]) / (h[k,j,i] − h[k,j+1,i])
        h_R[k,j+½,i] = h[k,j+1,i] + ½ φ(r) (h[k,j,i]   − h[k,j+1,i])
    Falls back to 1st-order upwind on north boundary where j+2 unavailable.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Scalar at T-points.
    v : Float[Array, "Nz Ny Nx"]
        y-velocity at V-points.
    limiter : str
        Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
        or ``'mc'``.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        North-face flux with zero ghost ring.
    """
    phi = _get_limiter(limiter)
    # Positive flow: h_face = h[k,j,i] + 0.5*phi(r)*(h[k,j+1,i]-h[k,j,i])
    diff_fwd = h[1:-1, 2:, 1:-1] - h[1:-1, 1:-1, 1:-1]
    diff_bwd = h[1:-1, 1:-1, 1:-1] - h[1:-1, :-2, 1:-1]
    r_pos = diff_bwd / (diff_fwd + _TVD_EPS)
    h_pos = h[1:-1, 1:-1, 1:-1] + 0.5 * phi(r_pos) * diff_fwd
    # Negative flow: h_face = h[k,j+1,i] + 0.5*phi(r)*(h[k,j,i]-h[k,j+1,i])
    diff_neg = h[1:-1, 1:-2, 1:-1] - h[1:-1, 2:-1, 1:-1]
    diff_neg2 = h[1:-1, 2:-1, 1:-1] - h[1:-1, 3:, 1:-1]
    r_neg_int = diff_neg2 / (diff_neg + _TVD_EPS)
    h_neg_interior = h[1:-1, 2:-1, 1:-1] + 0.5 * phi(r_neg_int) * diff_neg
    # 1st-order upwind fallback at north boundary row (j=Ny-2)
    h_neg_boundary = h[1:-1, -1:, 1:-1]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
    h_face = jnp.where(v[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
    return out

tvd_y_masked(h, v, mask, limiter='minmod')

TVD north-face flux over all z-levels with mask-aware stencil selection.

The 2-D mask is broadcast over the z-dimension. Falls back to 1st-order upwind near coastlines where the TVD stencil crosses land.

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Cell-centre tracer field.

 required
v Float[Array, 'Nz Ny Nx']

North-face velocity.

 required
mask ArakawaCGridMask

2-D Arakawa C-grid mask (broadcast over z).

 required
limiter str

Flux limiter name: 'minmod', 'van_leer', 'superbee', or 'mc'.

 'minmod'

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

North-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
def tvd_y_masked(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
    mask: ArakawaCGridMask,
    limiter: str = "minmod",
) -> Float[Array, "Nz Ny Nx"]:
    """TVD north-face flux over all z-levels with mask-aware stencil selection.

    The 2-D ``mask`` is broadcast over the z-dimension.  Falls back to
    1st-order upwind near coastlines where the TVD stencil crosses land.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Cell-centre tracer field.
    v : Float[Array, "Nz Ny Nx"]
        North-face velocity.
    mask : ArakawaCGridMask
        2-D Arakawa C-grid mask (broadcast over z).
    limiter : str
        Flux limiter name: ``'minmod'``, ``'van_leer'``, ``'superbee'``,
        or ``'mc'``.

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        North-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4))
    m_tvd = amasks[4]
    fn_tvd = self.tvd_y(h, v, limiter=limiter)
    fn_u1 = self.upwind1_y(h, v)
    pos_flow = v[1:-1, 1:-1, 1:-1] >= 0.0
    use_tvd = jnp.where(pos_flow, m_tvd[None, 1:-1, 1:-1], m_tvd[None, 2:, 1:-1])
    selected = jnp.where(use_tvd, fn_tvd[1:-1, 1:-1, 1:-1], fn_u1[1:-1, 1:-1, 1:-1])
    out = interior(selected, h)
    return out

upwind1_x(h, u)

1st-order upwind east-face flux over all z-levels.

Source code in finitevolx/_src/advection/reconstruction.py 
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
def upwind1_x(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """1st-order upwind east-face flux over all z-levels."""
    h_face = jnp.where(
        u[1:-1, 1:-1, 1:-1] >= 0.0,
        h[1:-1, 1:-1, 1:-1],
        h[1:-1, 1:-1, 2:],
    )
    out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
    return out

upwind1_y(h, v)

1st-order upwind north-face flux over all z-levels.

Source code in finitevolx/_src/advection/reconstruction.py 
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
def upwind1_y(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """1st-order upwind north-face flux over all z-levels."""
    h_face = jnp.where(
        v[1:-1, 1:-1, 1:-1] >= 0.0,
        h[1:-1, 1:-1, 1:-1],
        h[1:-1, 2:, 1:-1],
    )
    out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
    return out

weno3_x(h, u)

3-point WENO east-face flux over all z-levels with boundary fallback.

Positive flow: fe[k,j,i+1/2] = WENO3(h[k,j,i-1], h[k,j,i], h[k,j,i+1]) * u Negative flow: fe[k,j,i+1/2] = WENO3_right(h[k,j,i], h[k,j,i+1], h[k,j,i+2]) * u Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
def weno3_x(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """3-point WENO east-face flux over all z-levels with boundary fallback.

    Positive flow:  fe[k,j,i+1/2] = WENO3(h[k,j,i-1], h[k,j,i],   h[k,j,i+1]) * u
    Negative flow:  fe[k,j,i+1/2] = WENO3_right(h[k,j,i], h[k,j,i+1], h[k,j,i+2]) * u
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.
    """
    # WENO-3 positive: left-biased, valid for all interior faces
    h_pos = _weno3(h[1:-1, 1:-1, :-2], h[1:-1, 1:-1, 1:-1], h[1:-1, 1:-1, 2:])
    # WENO-3 negative: right-biased, valid for i+2 < Nx
    h_neg_interior = _weno3_right(
        h[1:-1, 1:-1, 1:-2], h[1:-1, 1:-1, 2:-1], h[1:-1, 1:-1, 3:]
    )
    # 1st-order upwind fallback at east boundary
    h_neg_boundary = h[1:-1, 1:-1, -1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=2)
    h_face = jnp.where(u[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
    return out

weno3_x_masked(h, u, mask)

3-point WENO east-face flux over all z-levels with mask-aware stencil selection.

The 2-D mask is broadcast over the z-dimension. Falls back to 1st-order upwind near coastlines where the WENO3 stencil crosses land.

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Cell-centre tracer field.

 required
u Float[Array, 'Nz Ny Nx']

East-face velocity.

 required
mask ArakawaCGridMask

2-D Arakawa C-grid mask (broadcast over z).

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

East-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
def weno3_x_masked(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
    mask: ArakawaCGridMask,
) -> Float[Array, "Nz Ny Nx"]:
    """3-point WENO east-face flux over all z-levels with mask-aware stencil selection.

    The 2-D ``mask`` is broadcast over the z-dimension.  Falls back to
    1st-order upwind near coastlines where the WENO3 stencil crosses land.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Cell-centre tracer field.
    u : Float[Array, "Nz Ny Nx"]
        East-face velocity.
    mask : ArakawaCGridMask
        2-D Arakawa C-grid mask (broadcast over z).

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        East-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4))
    m_w3 = amasks[4]
    fe_w3 = self.weno3_x(h, u)
    fe_u1 = self.upwind1_x(h, u)
    pos_flow = u[1:-1, 1:-1, 1:-1] >= 0.0
    use_w3 = jnp.where(pos_flow, m_w3[None, 1:-1, 1:-1], m_w3[None, 1:-1, 2:])
    selected = jnp.where(use_w3, fe_w3[1:-1, 1:-1, 1:-1], fe_u1[1:-1, 1:-1, 1:-1])
    out = interior(selected, h)
    return out

weno3_y(h, v)

3-point WENO north-face flux over all z-levels with boundary fallback.

Positive flow: fn[k,j+1/2,i] = WENO3(h[k,j-1,i], h[k,j,i], h[k,j+1,i]) * v Negative flow: fn[k,j+1/2,i] = WENO3_right(h[k,j,i], h[k,j+1,i], h[k,j+2,i]) * v Falls back to 1st-order upwind on north boundary where j+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
def weno3_y(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """3-point WENO north-face flux over all z-levels with boundary fallback.

    Positive flow:  fn[k,j+1/2,i] = WENO3(h[k,j-1,i], h[k,j,i],   h[k,j+1,i]) * v
    Negative flow:  fn[k,j+1/2,i] = WENO3_right(h[k,j,i], h[k,j+1,i], h[k,j+2,i]) * v
    Falls back to 1st-order upwind on north boundary where j+2 unavailable.
    """
    # WENO-3 positive: left-biased, valid for all interior faces
    h_pos = _weno3(h[1:-1, :-2, 1:-1], h[1:-1, 1:-1, 1:-1], h[1:-1, 2:, 1:-1])
    # WENO-3 negative: right-biased, valid for j+2 < Ny
    h_neg_interior = _weno3_right(
        h[1:-1, 1:-2, 1:-1], h[1:-1, 2:-1, 1:-1], h[1:-1, 3:, 1:-1]
    )
    # 1st-order upwind fallback at north boundary
    h_neg_boundary = h[1:-1, -1:, 1:-1]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
    h_face = jnp.where(v[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
    return out

weno3_y_masked(h, v, mask)

3-point WENO north-face flux over all z-levels with mask-aware stencil selection.

The 2-D mask is broadcast over the z-dimension. Falls back to 1st-order upwind near coastlines where the WENO3 stencil crosses land.

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Cell-centre tracer field.

 required
v Float[Array, 'Nz Ny Nx']

North-face velocity.

 required
mask ArakawaCGridMask

2-D Arakawa C-grid mask (broadcast over z).

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

North-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
def weno3_y_masked(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
    mask: ArakawaCGridMask,
) -> Float[Array, "Nz Ny Nx"]:
    """3-point WENO north-face flux over all z-levels with mask-aware stencil selection.

    The 2-D ``mask`` is broadcast over the z-dimension.  Falls back to
    1st-order upwind near coastlines where the WENO3 stencil crosses land.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Cell-centre tracer field.
    v : Float[Array, "Nz Ny Nx"]
        North-face velocity.
    mask : ArakawaCGridMask
        2-D Arakawa C-grid mask (broadcast over z).

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        North-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4))
    m_w3 = amasks[4]
    fn_w3 = self.weno3_y(h, v)
    fn_u1 = self.upwind1_y(h, v)
    pos_flow = v[1:-1, 1:-1, 1:-1] >= 0.0
    use_w3 = jnp.where(pos_flow, m_w3[None, 1:-1, 1:-1], m_w3[None, 2:, 1:-1])
    selected = jnp.where(use_w3, fn_w3[1:-1, 1:-1, 1:-1], fn_u1[1:-1, 1:-1, 1:-1])
    out = interior(selected, h)
    return out

weno5_x(h, u)

5-point WENO east-face flux over all z-levels with sign-dependent fallbacks.

Positive flow: fe[k,j,i+1/2] = WENO5(h[k,j,i-2..i+2]) * u for i = 2..Nx-3; WENO3 fallback at i = 1 and i = Nx-2. Negative flow: fe[k,j,i+1/2] = WENO5_right(h[k,j,i-1], h[k,j,i], h[k,j,i+1], h[k,j,i+2], h[k,j,i+3]) * u for i = 1..Nx-4; WENO3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.

Source code in finitevolx/_src/advection/reconstruction.py 
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
def weno5_x(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """5-point WENO east-face flux over all z-levels with sign-dependent fallbacks.

    Positive flow:  fe[k,j,i+1/2] = WENO5(h[k,j,i-2..i+2]) * u
        for i = 2..Nx-3; WENO3 fallback at i = 1 and i = Nx-2.
    Negative flow:  fe[k,j,i+1/2] = WENO5_right(h[k,j,i-1], h[k,j,i], h[k,j,i+1], h[k,j,i+2], h[k,j,i+3]) * u
        for i = 1..Nx-4; WENO3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
    """
    h5_pos_int = _weno5(
        h[1:-1, 1:-1, :-4],
        h[1:-1, 1:-1, 1:-3],
        h[1:-1, 1:-1, 2:-2],
        h[1:-1, 1:-1, 3:-1],
        h[1:-1, 1:-1, 4:],
    )
    h3_pos_first = _weno3(
        h[1:-1, 1:-1, 0:1], h[1:-1, 1:-1, 1:2], h[1:-1, 1:-1, 2:3]
    )
    h3_pos_last = _weno3(
        h[1:-1, 1:-1, -3:-2], h[1:-1, 1:-1, -2:-1], h[1:-1, 1:-1, -1:]
    )
    h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=2)
    h5_neg_int = _weno5_right(
        h[1:-1, 1:-1, :-4],
        h[1:-1, 1:-1, 1:-3],
        h[1:-1, 1:-1, 2:-2],
        h[1:-1, 1:-1, 3:-1],
        h[1:-1, 1:-1, 4:],
    )
    h3_neg_penultimate = _weno3_right(
        h[1:-1, 1:-1, -3:-2], h[1:-1, 1:-1, -2:-1], h[1:-1, 1:-1, -1:]
    )
    h_neg = jnp.concatenate(
        [h5_neg_int, h3_neg_penultimate, h[1:-1, 1:-1, -1:]], axis=2
    )
    h_face = jnp.where(u[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
    return out

weno5_x_masked(h, u, mask)

5-point WENO east-face flux over all z-levels with mask-aware adaptive stencil.

The 2-D mask is broadcast over the z-dimension so that the same horizontal stencil-capability map is applied at every depth level. Adaptively falls back to WENO3 or 1st-order upwind near coastlines.

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Cell-centre tracer field.

 required
u Float[Array, 'Nz Ny Nx']

East-face velocity.

 required
mask ArakawaCGridMask

2-D Arakawa C-grid mask (broadcast over z).

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

East-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
def weno5_x_masked(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
    mask: ArakawaCGridMask,
) -> Float[Array, "Nz Ny Nx"]:
    """5-point WENO east-face flux over all z-levels with mask-aware adaptive stencil.

    The 2-D ``mask`` is broadcast over the z-dimension so that the same
    horizontal stencil-capability map is applied at every depth level.
    Adaptively falls back to WENO3 or 1st-order upwind near coastlines.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Cell-centre tracer field.
    u : Float[Array, "Nz Ny Nx"]
        East-face velocity.
    mask : ArakawaCGridMask
        2-D Arakawa C-grid mask (broadcast over z).

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        East-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4, 6))
    m5 = amasks[6]  # (Ny, Nx)
    m3 = amasks[4]
    fe_w5 = self.weno5_x(h, u)
    fe_w3 = self.weno3_x(h, u)
    fe_u1 = self.upwind1_x(h, u)
    # Select tier based on upwind-cell stencil capability, broadcast mask over z
    # Positive flow: upwind cell is (j, i)   → mask at [1:-1, 1:-1]
    # Negative flow: upwind cell is (j, i+1) → mask at [1:-1, 2:]
    pos_flow = u[1:-1, 1:-1, 1:-1] >= 0.0
    use_w5 = jnp.where(pos_flow, m5[None, 1:-1, 1:-1], m5[None, 1:-1, 2:])
    use_w3 = jnp.where(pos_flow, m3[None, 1:-1, 1:-1], m3[None, 1:-1, 2:])
    selected = jnp.where(
        use_w5,
        fe_w5[1:-1, 1:-1, 1:-1],
        jnp.where(use_w3, fe_w3[1:-1, 1:-1, 1:-1], fe_u1[1:-1, 1:-1, 1:-1]),
    )
    out = interior(selected, h)
    return out

weno5_y(h, v)

5-point WENO north-face flux over all z-levels with sign-dependent fallbacks.

Positive flow: fn[k,j+1/2,i] = WENO5(h[k,j-2..j+2,i]) * v for j = 2..Ny-3; WENO3 fallback at j = 1 and j = Ny-2. Negative flow: fn[k,j+1/2,i] = WENO5_right(h[k,j-1,i], h[k,j,i], h[k,j+1,i], h[k,j+2,i], h[k,j+3,i]) * v for j = 1..Ny-4; WENO3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.

Source code in finitevolx/_src/advection/reconstruction.py 
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
def weno5_y(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """5-point WENO north-face flux over all z-levels with sign-dependent fallbacks.

    Positive flow:  fn[k,j+1/2,i] = WENO5(h[k,j-2..j+2,i]) * v
        for j = 2..Ny-3; WENO3 fallback at j = 1 and j = Ny-2.
    Negative flow:  fn[k,j+1/2,i] = WENO5_right(h[k,j-1,i], h[k,j,i], h[k,j+1,i], h[k,j+2,i], h[k,j+3,i]) * v
        for j = 1..Ny-4; WENO3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.
    """
    h5_pos_int = _weno5(
        h[1:-1, :-4, 1:-1],
        h[1:-1, 1:-3, 1:-1],
        h[1:-1, 2:-2, 1:-1],
        h[1:-1, 3:-1, 1:-1],
        h[1:-1, 4:, 1:-1],
    )
    h3_pos_first = _weno3(
        h[1:-1, 0:1, 1:-1], h[1:-1, 1:2, 1:-1], h[1:-1, 2:3, 1:-1]
    )
    h3_pos_last = _weno3(
        h[1:-1, -3:-2, 1:-1], h[1:-1, -2:-1, 1:-1], h[1:-1, -1:, 1:-1]
    )
    h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=1)
    h5_neg_int = _weno5_right(
        h[1:-1, :-4, 1:-1],
        h[1:-1, 1:-3, 1:-1],
        h[1:-1, 2:-2, 1:-1],
        h[1:-1, 3:-1, 1:-1],
        h[1:-1, 4:, 1:-1],
    )
    h3_neg_penultimate = _weno3_right(
        h[1:-1, -3:-2, 1:-1], h[1:-1, -2:-1, 1:-1], h[1:-1, -1:, 1:-1]
    )
    h_neg = jnp.concatenate(
        [h5_neg_int, h3_neg_penultimate, h[1:-1, -1:, 1:-1]], axis=1
    )
    h_face = jnp.where(v[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
    return out

weno5_y_masked(h, v, mask)

5-point WENO north-face flux over all z-levels with mask-aware adaptive stencil.

The 2-D mask is broadcast over the z-dimension so that the same horizontal stencil-capability map is applied at every depth level. Adaptively falls back to WENO3 or 1st-order upwind near coastlines.

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Cell-centre tracer field.

 required
v Float[Array, 'Nz Ny Nx']

North-face velocity.

 required
mask ArakawaCGridMask

2-D Arakawa C-grid mask (broadcast over z).

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

North-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
def weno5_y_masked(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
    mask: ArakawaCGridMask,
) -> Float[Array, "Nz Ny Nx"]:
    """5-point WENO north-face flux over all z-levels with mask-aware adaptive stencil.

    The 2-D ``mask`` is broadcast over the z-dimension so that the same
    horizontal stencil-capability map is applied at every depth level.
    Adaptively falls back to WENO3 or 1st-order upwind near coastlines.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Cell-centre tracer field.
    v : Float[Array, "Nz Ny Nx"]
        North-face velocity.
    mask : ArakawaCGridMask
        2-D Arakawa C-grid mask (broadcast over z).

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        North-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4, 6))
    m5 = amasks[6]
    m3 = amasks[4]
    fn_w5 = self.weno5_y(h, v)
    fn_w3 = self.weno3_y(h, v)
    fn_u1 = self.upwind1_y(h, v)
    # Select tier based on upwind-cell stencil capability, broadcast mask over z
    # Positive flow: upwind cell is (j, i)   → mask at [1:-1, 1:-1]
    # Negative flow: upwind cell is (j+1, i) → mask at [2:, 1:-1]
    pos_flow = v[1:-1, 1:-1, 1:-1] >= 0.0
    use_w5 = jnp.where(pos_flow, m5[None, 1:-1, 1:-1], m5[None, 2:, 1:-1])
    use_w3 = jnp.where(pos_flow, m3[None, 1:-1, 1:-1], m3[None, 2:, 1:-1])
    selected = jnp.where(
        use_w5,
        fn_w5[1:-1, 1:-1, 1:-1],
        jnp.where(use_w3, fn_w3[1:-1, 1:-1, 1:-1], fn_u1[1:-1, 1:-1, 1:-1]),
    )
    out = interior(selected, h)
    return out

weno7_x(h, u)

7-point WENO east-face flux over all z-levels with hierarchical fallbacks.

Source code in finitevolx/_src/advection/reconstruction.py 
1641
1642
1643
1644
1645
1646
1647
def weno7_x(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """7-point WENO east-face flux over all z-levels with hierarchical fallbacks."""
    return _weno_flux_axis_3d_x(h, u, order=7)

weno7_y(h, v)

7-point WENO north-face flux over all z-levels with hierarchical fallbacks.

Source code in finitevolx/_src/advection/reconstruction.py 
1649
1650
1651
1652
1653
1654
1655
def weno7_y(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """7-point WENO north-face flux over all z-levels with hierarchical fallbacks."""
    return _weno_flux_axis_3d_y(h, v, order=7)

weno9_x(h, u)

9-point WENO east-face flux over all z-levels with hierarchical fallbacks.

Source code in finitevolx/_src/advection/reconstruction.py 
1657
1658
1659
1660
1661
1662
1663
def weno9_x(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """9-point WENO east-face flux over all z-levels with hierarchical fallbacks."""
    return _weno_flux_axis_3d_x(h, u, order=9)

weno9_y(h, v)

9-point WENO north-face flux over all z-levels with hierarchical fallbacks.

Source code in finitevolx/_src/advection/reconstruction.py 
1665
1666
1667
1668
1669
1670
1671
def weno9_y(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """9-point WENO north-face flux over all z-levels with hierarchical fallbacks."""
    return _weno_flux_axis_3d_y(h, v, order=9)

wenoz3_x(h, u)

3-point WENO-Z east-face flux over all z-levels with boundary fallback.

Positive flow: fe[k,j,i+1/2] = WENOZ3(h[k,j,i-1], h[k,j,i], h[k,j,i+1]) * u Negative flow: fe[k,j,i+1/2] = WENOZ3_right(h[k,j,i], h[k,j,i+1], h[k,j,i+2]) * u Falls back to 1st-order upwind on east boundary where i+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
def wenoz3_x(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """3-point WENO-Z east-face flux over all z-levels with boundary fallback.

    Positive flow:  fe[k,j,i+1/2] = WENOZ3(h[k,j,i-1], h[k,j,i],   h[k,j,i+1]) * u
    Negative flow:  fe[k,j,i+1/2] = WENOZ3_right(h[k,j,i], h[k,j,i+1], h[k,j,i+2]) * u
    Falls back to 1st-order upwind on east boundary where i+2 unavailable.
    """
    # WENO-Z-3 positive: left-biased, valid for all interior faces
    h_pos = _wenoz3(h[1:-1, 1:-1, :-2], h[1:-1, 1:-1, 1:-1], h[1:-1, 1:-1, 2:])
    # WENO-Z-3 negative: right-biased, valid for i+2 < Nx
    h_neg_interior = _wenoz3_right(
        h[1:-1, 1:-1, 1:-2], h[1:-1, 1:-1, 2:-1], h[1:-1, 1:-1, 3:]
    )
    # 1st-order upwind fallback at east boundary
    h_neg_boundary = h[1:-1, 1:-1, -1:]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=2)
    h_face = jnp.where(u[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
    return out

wenoz3_y(h, v)

3-point WENO-Z north-face flux over all z-levels with boundary fallback.

Positive flow: fn[k,j+1/2,i] = WENOZ3(h[k,j-1,i], h[k,j,i], h[k,j+1,i]) * v Negative flow: fn[k,j+1/2,i] = WENOZ3_right(h[k,j,i], h[k,j+1,i], h[k,j+2,i]) * v Falls back to 1st-order upwind on north boundary where j+2 unavailable.

Source code in finitevolx/_src/advection/reconstruction.py 
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
def wenoz3_y(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """3-point WENO-Z north-face flux over all z-levels with boundary fallback.

    Positive flow:  fn[k,j+1/2,i] = WENOZ3(h[k,j-1,i], h[k,j,i],   h[k,j+1,i]) * v
    Negative flow:  fn[k,j+1/2,i] = WENOZ3_right(h[k,j,i], h[k,j+1,i], h[k,j+2,i]) * v
    Falls back to 1st-order upwind on north boundary where j+2 unavailable.
    """
    # WENO-Z-3 positive: left-biased, valid for all interior faces
    h_pos = _wenoz3(h[1:-1, :-2, 1:-1], h[1:-1, 1:-1, 1:-1], h[1:-1, 2:, 1:-1])
    # WENO-Z-3 negative: right-biased, valid for j+2 < Ny
    h_neg_interior = _wenoz3_right(
        h[1:-1, 1:-2, 1:-1], h[1:-1, 2:-1, 1:-1], h[1:-1, 3:, 1:-1]
    )
    # 1st-order upwind fallback at north boundary
    h_neg_boundary = h[1:-1, -1:, 1:-1]
    h_neg = jnp.concatenate([h_neg_interior, h_neg_boundary], axis=1)
    h_face = jnp.where(v[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
    return out

wenoz5_x(h, u)

5-point WENO-Z east-face flux over all z-levels with sign-dependent fallbacks.

Positive flow: fe[k,j,i+1/2] = WENOZ5(h[k,j,i-2..i+2]) * u for i = 2..Nx-3; WENO-Z-3 fallback at i = 1 and i = Nx-2. Negative flow: fe[k,j,i+1/2] = WENOZ5_right(h[k,j,i-1], h[k,j,i], h[k,j,i+1], h[k,j,i+2], h[k,j,i+3]) * u for i = 1..Nx-4; WENO-Z-3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.

Source code in finitevolx/_src/advection/reconstruction.py 
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
def wenoz5_x(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """5-point WENO-Z east-face flux over all z-levels with sign-dependent fallbacks.

    Positive flow:  fe[k,j,i+1/2] = WENOZ5(h[k,j,i-2..i+2]) * u
        for i = 2..Nx-3; WENO-Z-3 fallback at i = 1 and i = Nx-2.
    Negative flow:  fe[k,j,i+1/2] = WENOZ5_right(h[k,j,i-1], h[k,j,i], h[k,j,i+1], h[k,j,i+2], h[k,j,i+3]) * u
        for i = 1..Nx-4; WENO-Z-3_right fallback at i = Nx-3; 1st-order upwind at i = Nx-2.
    """
    h5_pos_int = _wenoz5(
        h[1:-1, 1:-1, :-4],
        h[1:-1, 1:-1, 1:-3],
        h[1:-1, 1:-1, 2:-2],
        h[1:-1, 1:-1, 3:-1],
        h[1:-1, 1:-1, 4:],
    )
    h3_pos_first = _wenoz3(
        h[1:-1, 1:-1, 0:1], h[1:-1, 1:-1, 1:2], h[1:-1, 1:-1, 2:3]
    )
    h3_pos_last = _wenoz3(
        h[1:-1, 1:-1, -3:-2], h[1:-1, 1:-1, -2:-1], h[1:-1, 1:-1, -1:]
    )
    h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=2)
    h5_neg_int = _wenoz5_right(
        h[1:-1, 1:-1, :-4],
        h[1:-1, 1:-1, 1:-3],
        h[1:-1, 1:-1, 2:-2],
        h[1:-1, 1:-1, 3:-1],
        h[1:-1, 1:-1, 4:],
    )
    h3_neg_penultimate = _wenoz3_right(
        h[1:-1, 1:-1, -3:-2], h[1:-1, 1:-1, -2:-1], h[1:-1, 1:-1, -1:]
    )
    h_neg = jnp.concatenate(
        [h5_neg_int, h3_neg_penultimate, h[1:-1, 1:-1, -1:]], axis=2
    )
    h_face = jnp.where(u[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * u[1:-1, 1:-1, 1:-1], h)
    return out

wenoz5_x_masked(h, u, mask)

5-point WENO-Z east-face flux over all z-levels with mask-aware adaptive stencil.

The 2-D mask is broadcast over the z-dimension so that the same horizontal stencil-capability map is applied at every depth level. Adaptively falls back to WENO-Z-3 or 1st-order upwind near coastlines.

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Cell-centre tracer field.

 required
u Float[Array, 'Nz Ny Nx']

East-face velocity.

 required
mask ArakawaCGridMask

2-D Arakawa C-grid mask (broadcast over z).

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

East-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
def wenoz5_x_masked(
    self,
    h: Float[Array, "Nz Ny Nx"],
    u: Float[Array, "Nz Ny Nx"],
    mask: ArakawaCGridMask,
) -> Float[Array, "Nz Ny Nx"]:
    """5-point WENO-Z east-face flux over all z-levels with mask-aware adaptive stencil.

    The 2-D ``mask`` is broadcast over the z-dimension so that the same
    horizontal stencil-capability map is applied at every depth level.
    Adaptively falls back to WENO-Z-3 or 1st-order upwind near coastlines.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Cell-centre tracer field.
    u : Float[Array, "Nz Ny Nx"]
        East-face velocity.
    mask : ArakawaCGridMask
        2-D Arakawa C-grid mask (broadcast over z).

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        East-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="x", stencil_sizes=(2, 4, 6))
    m5 = amasks[6]
    m3 = amasks[4]
    fe_w5 = self.wenoz5_x(h, u)
    fe_w3 = self.wenoz3_x(h, u)
    fe_u1 = self.upwind1_x(h, u)
    pos_flow = u[1:-1, 1:-1, 1:-1] >= 0.0
    use_w5 = jnp.where(pos_flow, m5[None, 1:-1, 1:-1], m5[None, 1:-1, 2:])
    use_w3 = jnp.where(pos_flow, m3[None, 1:-1, 1:-1], m3[None, 1:-1, 2:])
    selected = jnp.where(
        use_w5,
        fe_w5[1:-1, 1:-1, 1:-1],
        jnp.where(use_w3, fe_w3[1:-1, 1:-1, 1:-1], fe_u1[1:-1, 1:-1, 1:-1]),
    )
    out = interior(selected, h)
    return out

wenoz5_y(h, v)

5-point WENO-Z north-face flux over all z-levels with sign-dependent fallbacks.

Positive flow: fn[k,j+1/2,i] = WENOZ5(h[k,j-2..j+2,i]) * v for j = 2..Ny-3; WENO-Z-3 fallback at j = 1 and j = Ny-2. Negative flow: fn[k,j+1/2,i] = WENOZ5_right(h[k,j-1,i], h[k,j,i], h[k,j+1,i], h[k,j+2,i], h[k,j+3,i]) * v for j = 1..Ny-4; WENO-Z-3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.

Source code in finitevolx/_src/advection/reconstruction.py 
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
def wenoz5_y(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
) -> Float[Array, "Nz Ny Nx"]:
    """5-point WENO-Z north-face flux over all z-levels with sign-dependent fallbacks.

    Positive flow:  fn[k,j+1/2,i] = WENOZ5(h[k,j-2..j+2,i]) * v
        for j = 2..Ny-3; WENO-Z-3 fallback at j = 1 and j = Ny-2.
    Negative flow:  fn[k,j+1/2,i] = WENOZ5_right(h[k,j-1,i], h[k,j,i], h[k,j+1,i], h[k,j+2,i], h[k,j+3,i]) * v
        for j = 1..Ny-4; WENO-Z-3_right fallback at j = Ny-3; 1st-order upwind at j = Ny-2.
    """
    h5_pos_int = _wenoz5(
        h[1:-1, :-4, 1:-1],
        h[1:-1, 1:-3, 1:-1],
        h[1:-1, 2:-2, 1:-1],
        h[1:-1, 3:-1, 1:-1],
        h[1:-1, 4:, 1:-1],
    )
    h3_pos_first = _wenoz3(
        h[1:-1, 0:1, 1:-1], h[1:-1, 1:2, 1:-1], h[1:-1, 2:3, 1:-1]
    )
    h3_pos_last = _wenoz3(
        h[1:-1, -3:-2, 1:-1], h[1:-1, -2:-1, 1:-1], h[1:-1, -1:, 1:-1]
    )
    h_pos = jnp.concatenate([h3_pos_first, h5_pos_int, h3_pos_last], axis=1)
    h5_neg_int = _wenoz5_right(
        h[1:-1, :-4, 1:-1],
        h[1:-1, 1:-3, 1:-1],
        h[1:-1, 2:-2, 1:-1],
        h[1:-1, 3:-1, 1:-1],
        h[1:-1, 4:, 1:-1],
    )
    h3_neg_penultimate = _wenoz3_right(
        h[1:-1, -3:-2, 1:-1], h[1:-1, -2:-1, 1:-1], h[1:-1, -1:, 1:-1]
    )
    h_neg = jnp.concatenate(
        [h5_neg_int, h3_neg_penultimate, h[1:-1, -1:, 1:-1]], axis=1
    )
    h_face = jnp.where(v[1:-1, 1:-1, 1:-1] >= 0.0, h_pos, h_neg)
    out = interior(h_face * v[1:-1, 1:-1, 1:-1], h)
    return out

wenoz5_y_masked(h, v, mask)

5-point WENO-Z north-face flux over all z-levels with mask-aware adaptive stencil.

The 2-D mask is broadcast over the z-dimension so that the same horizontal stencil-capability map is applied at every depth level. Adaptively falls back to WENO-Z-3 or 1st-order upwind near coastlines.

Parameters:

Name Type Description Default
h Float[Array, 'Nz Ny Nx']

Cell-centre tracer field.

 required
v Float[Array, 'Nz Ny Nx']

North-face velocity.

 required
mask ArakawaCGridMask

2-D Arakawa C-grid mask (broadcast over z).

 required

Returns:

Type Description
Float[Array, 'Nz Ny Nx']

North-face flux with zero ghost ring.
Source code in finitevolx/_src/advection/reconstruction.py 
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
def wenoz5_y_masked(
    self,
    h: Float[Array, "Nz Ny Nx"],
    v: Float[Array, "Nz Ny Nx"],
    mask: ArakawaCGridMask,
) -> Float[Array, "Nz Ny Nx"]:
    """5-point WENO-Z north-face flux over all z-levels with mask-aware adaptive stencil.

    The 2-D ``mask`` is broadcast over the z-dimension so that the same
    horizontal stencil-capability map is applied at every depth level.
    Adaptively falls back to WENO-Z-3 or 1st-order upwind near coastlines.

    Parameters
    ----------
    h : Float[Array, "Nz Ny Nx"]
        Cell-centre tracer field.
    v : Float[Array, "Nz Ny Nx"]
        North-face velocity.
    mask : ArakawaCGridMask
        2-D Arakawa C-grid mask (broadcast over z).

    Returns
    -------
    Float[Array, "Nz Ny Nx"]
        North-face flux with zero ghost ring.
    """
    amasks = mask.get_adaptive_masks(direction="y", stencil_sizes=(2, 4, 6))
    m5 = amasks[6]
    m3 = amasks[4]
    fn_w5 = self.wenoz5_y(h, v)
    fn_w3 = self.wenoz3_y(h, v)
    fn_u1 = self.upwind1_y(h, v)
    pos_flow = v[1:-1, 1:-1, 1:-1] >= 0.0
    use_w5 = jnp.where(pos_flow, m5[None, 1:-1, 1:-1], m5[None, 2:, 1:-1])
    use_w3 = jnp.where(pos_flow, m3[None, 1:-1, 1:-1], m3[None, 2:, 1:-1])
    selected = jnp.where(
        use_w5,
        fn_w5[1:-1, 1:-1, 1:-1],
        jnp.where(use_w3, fn_w3[1:-1, 1:-1, 1:-1], fn_u1[1:-1, 1:-1, 1:-1]),
    )
    out = interior(selected, h)
    return out

Flux Limiters

finitevolx.minmod(r)

Minmod flux limiter.

φ(r) = max(0, min(1, r))

The most diffusive TVD limiter; uses only the minimum of the two consecutive slopes. Equivalent to the most restrictive TVD limiter.

Parameters:

Name Type Description Default
r array

Ratio of consecutive upwind differences.

 required

Returns:

Type Description
array

Limiter values φ(r) ∈ [0, 1].
Source code in finitevolx/_src/advection/limiters.py 
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
def minmod(r: Float[Array, ...]) -> Float[Array, ...]:
    """Minmod flux limiter.

    φ(r) = max(0, min(1, r))

    The most diffusive TVD limiter; uses only the minimum of the two
    consecutive slopes.  Equivalent to the most restrictive TVD limiter.

    Parameters
    ----------
    r : array
        Ratio of consecutive upwind differences.

    Returns
    -------
    array
        Limiter values φ(r) ∈ [0, 1].
    """
    return jnp.maximum(0.0, jnp.minimum(1.0, r))

finitevolx.mc(r)

Monotonized Central (MC) flux limiter.

φ(r) = max(0, min((1 + r)/2, 2r, 2))

A second-order accurate limiter that is less compressive than Superbee and smoother than Minmod. Sometimes called the MC limiter or Monotonized Central-difference limiter.

Parameters:

Name Type Description Default
r array

Ratio of consecutive upwind differences.

 required

Returns:

Type Description
array

Limiter values φ(r) ∈ [0, 2].
Source code in finitevolx/_src/advection/limiters.py 
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
def mc(r: Float[Array, ...]) -> Float[Array, ...]:
    """Monotonized Central (MC) flux limiter.

    φ(r) = max(0, min((1 + r)/2, 2r, 2))

    A second-order accurate limiter that is less compressive than Superbee
    and smoother than Minmod.  Sometimes called the MC limiter or
    Monotonized Central-difference limiter.

    Parameters
    ----------
    r : array
        Ratio of consecutive upwind differences.

    Returns
    -------
    array
        Limiter values φ(r) ∈ [0, 2].
    """
    return jnp.maximum(
        0.0,
        jnp.minimum(jnp.minimum((1.0 + r) / 2.0, 2.0 * r), 2.0),
    )

finitevolx.superbee(r)

Superbee flux limiter.

φ(r) = max(0, max(min(2r, 1), min(r, 2)))

The most compressive TVD limiter; produces the sharpest fronts but can introduce steepening artefacts on smooth solutions.

Parameters:

Name Type Description Default
r array

Ratio of consecutive upwind differences.

 required

Returns:

Type Description
array

Limiter values φ(r) ∈ [0, 2].
Source code in finitevolx/_src/advection/limiters.py 
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
def superbee(r: Float[Array, ...]) -> Float[Array, ...]:
    """Superbee flux limiter.

    φ(r) = max(0, max(min(2r, 1), min(r, 2)))

    The most compressive TVD limiter; produces the sharpest fronts but can
    introduce steepening artefacts on smooth solutions.

    Parameters
    ----------
    r : array
        Ratio of consecutive upwind differences.

    Returns
    -------
    array
        Limiter values φ(r) ∈ [0, 2].
    """
    return jnp.maximum(
        0.0,
        jnp.maximum(jnp.minimum(2.0 * r, 1.0), jnp.minimum(r, 2.0)),
    )

finitevolx.van_leer(r)

Van Leer flux limiter.

φ(r) = (r + |r|) / (1 + |r|)

A smooth limiter that is symmetric around r = 1. Returns 0 for r ≤ 0 and approaches 2 for r → ∞.

Parameters:

Name Type Description Default
r array

Ratio of consecutive upwind differences.

 required

Returns:

Type Description
array

Limiter values φ(r) ∈ [0, 2).
Source code in finitevolx/_src/advection/limiters.py 
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
def van_leer(r: Float[Array, ...]) -> Float[Array, ...]:
    """Van Leer flux limiter.

    φ(r) = (r + |r|) / (1 + |r|)

    A smooth limiter that is symmetric around r = 1.  Returns 0 for r ≤ 0
    and approaches 2 for r → ∞.

    Parameters
    ----------
    r : array
        Ratio of consecutive upwind differences.

    Returns
    -------
    array
        Limiter values φ(r) ∈ [0, 2).
    """
    r_abs = jnp.abs(r)
    return (r + r_abs) / (1.0 + r_abs)