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Fourier Grids

FourierGrid1D

Bases: Module

Fourier grid setup for periodic domain.

Mathematical Framework:

For periodic domain [0, L]:

Grid points: x_j = j·Δx = j·L/N, j = 0, 1, ..., N-1

Wavenumbers: k_n = 2πn/L, n = -N/2+1, ..., N/2
Discrete Fourier Transform

û_k = (1/N)·Σ_{j=0}^{N-1} u_j·exp(-ik·x_j)

u_j = Σ_{k=-N/2+1}^{N/2} û_k·exp(ik·x_j)

FFT Ordering

k = [0, 1, 2, ..., N/2-1, -N/2, -N/2+1, ..., -1]

N : int
    Number of grid points
L : float
    Domain length [m]
dx : float
    Grid spacing [m]. Must satisfy L = N * dx.
dealias : str
    Dealiasing method ('2/3', 'padding', None)
Source code in spectraldiffx/_src/fourier/grid.py 
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class FourierGrid1D(eqx.Module):
    """
    Fourier grid setup for periodic domain.

    Mathematical Framework:
    -----------------------
    For periodic domain [0, L]:

        Grid points: x_j = j·Δx = j·L/N, j = 0, 1, ..., N-1

        Wavenumbers: k_n = 2πn/L, n = -N/2+1, ..., N/2

    Discrete Fourier Transform:
        û_k = (1/N)·Σ_{j=0}^{N-1} u_j·exp(-ik·x_j)

        u_j = Σ_{k=-N/2+1}^{N/2} û_k·exp(ik·x_j)

    FFT Ordering:
        k = [0, 1, 2, ..., N/2-1, -N/2, -N/2+1, ..., -1]

    Attributes:
    -----------
        N : int
            Number of grid points
        L : float
            Domain length [m]
        dx : float
            Grid spacing [m]. Must satisfy L = N * dx.
        dealias : str
            Dealiasing method ('2/3', 'padding', None)
    """

    N: int
    L: float
    dx: float
    dealias: Literal["2/3", "padding", None] | None = "2/3"

    def check_consistency(self, rtol: float = 1e-5) -> bool:
        """
        Verifies that the grid attributes (N, L, dx) are mathematically consistent.

        Checks the relationship: L ≈ N * dx

        Parameters:
        -----------
        rtol : float
            Relative tolerance for the floating point comparison.

        Returns:
        --------
        bool
            True if consistent, raises ValueError otherwise.
        """
        expected_L = self.N * self.dx
        if not jnp.isclose(self.L, expected_L, rtol=rtol):
            raise ValueError(
                f"Grid inconsistency detected.\n"
                f"  L (defined): {self.L}\n"
                f"  N * dx     : {expected_L}\n"
                f"  Difference : {abs(self.L - expected_L)}"
            )
        return True

    @classmethod
    def from_N_L(
        cls,
        N: int,
        L: float,
        dealias: Literal["2/3", "padding", None] | None = "2/3",
    ) -> "FourierGrid1D":
        """
        Initialize FourierGrid using Number of points (N) and Length (L).

        Calculates: dx = L / N
        """
        dx = L / N
        return cls(N=N, L=L, dx=dx, dealias=dealias)

    @classmethod
    def from_N_dx(
        cls,
        N: int,
        dx: float,
        dealias: Literal["2/3", "padding", None] | None = "2/3",
    ) -> "FourierGrid1D":
        """
        Initialize FourierGrid using Number of points (N) and Spacing (dx).

        Calculates: L = N * dx
        """
        L = N * dx
        return cls(N=N, L=L, dx=dx, dealias=dealias)

    @classmethod
    def from_L_dx(
        cls,
        L: float,
        dx: float,
        dealias: Literal["2/3", "padding", None] | None = "2/3",
    ) -> "FourierGrid1D":
        """
        Initialize FourierGrid using Length (L) and Spacing (dx).

        Calculates: N = L / dx (Must result in an integer)

        Raises:
        -------
        ValueError: If L is not divisible by dx (N is not an integer).
        """
        N_float = L / dx
        if not jnp.isclose(N_float % 1, 0) and not jnp.isclose(N_float % 1, 1):
            raise ValueError(
                f"L={L} is not divisible by dx={dx}. Resulting N={N_float} is not an integer."
            )

        N = int(round(N_float))
        return cls(N=N, L=L, dx=dx, dealias=dealias)

    @property
    def x(self) -> Float[Array, "N"]:
        """Physical grid points: x_j = j·Δx"""
        # We use linspace for numerical stability, but it should match self.dx * j
        return jnp.linspace(0, self.L, self.N, endpoint=False)

    @property
    def k(self) -> Float[Array, "N"]:
        """
        Wavenumbers in FFT order.

        k = 2π·[0, 1, 2, ..., N/2-1, -N/2, -N/2+1, ..., -1]/L

        Returns:
        --------
        k : Array [N]
            Wavenumbers [rad/m]
        """
        return 2 * jnp.pi * jnp.fft.fftfreq(self.N, self.dx)

    @property
    def k_dealias(self) -> Float[Array, "N"]:
        """
        Dealiased wavenumbers (set high frequencies to zero).

        2/3 rule: Keep |k| ≤ k_max/3
        This prevents aliasing in quadratic nonlinearities.

        Mathematical Justification:
        ---------------------------
        For product w = u·v:
            ŵ_k = Σ_p û_p·v̂_{k-p}

        Maximum frequency in w: k_max(w) = k_max(u) + k_max(v)

        To avoid aliasing: k_max(w) ≤ N/2
        Therefore: 2·k_max(u) ≤ N/2  →  k_max(u) ≤ N/3

        Returns:
        --------
        k : Array [N]
            Dealiased wavenumbers
        """
        k = self.k
        if self.dealias == "2/3":
            k_max = jnp.abs(k).max()
            cutoff = k_max * 2 / 3
            k = jnp.where(jnp.abs(k) > cutoff, 0.0, k)
        return k

    def dealias_filter(self) -> Float[Array, "N"]:
        """
        Dealiasing filter: 1 for kept modes, 0 for removed.

        Returns:
        --------
        filter : Array [N]
            Filter mask (1 or 0)
        """
        if self.dealias == "2/3":
            k = self.k
            k_max = jnp.abs(k).max()
            cutoff = k_max * 2 / 3
            return jnp.where(jnp.abs(k) <= cutoff, 1.0, 0.0)
        else:
            return jnp.ones(self.N)

    def transform(self, u: Array, inverse: bool = False) -> Complex[Array, "N"]:
        """
        Perform 1D Fourier Transform.
        Forward: physical -> spectral [u -> u_hat]
        Inverse: spectral -> physical [u_hat -> u]
        """
        if inverse:
            return jnp.fft.ifft(u)
        return jnp.fft.fft(u)

Attributes

x property

Physical grid points: x_j = j·Δx

k property

Wavenumbers in FFT order.

k = 2π·[0, 1, 2, ..., N/2-1, -N/2, -N/2+1, ..., -1]/L

Returns:

k : Array [N] Wavenumbers [rad/m]

k_dealias property

Dealiased wavenumbers (set high frequencies to zero).

2/3 rule: Keep |k| ≤ k_max/3 This prevents aliasing in quadratic nonlinearities.

Mathematical Justification:

For product w = u·v: ŵ_k = Σ_p û_p·v̂_{k-p}

Maximum frequency in w: k_max(w) = k_max(u) + k_max(v)

To avoid aliasing: k_max(w) ≤ N/2 Therefore: 2·k_max(u) ≤ N/2 → k_max(u) ≤ N/3

Returns:

k : Array [N] Dealiased wavenumbers

Functions

check_consistency(rtol=1e-05)

Verifies that the grid attributes (N, L, dx) are mathematically consistent.

Checks the relationship: L ≈ N * dx

Parameters:

rtol : float Relative tolerance for the floating point comparison.

Returns:

bool True if consistent, raises ValueError otherwise.

Source code in spectraldiffx/_src/fourier/grid.py 
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def check_consistency(self, rtol: float = 1e-5) -> bool:
    """
    Verifies that the grid attributes (N, L, dx) are mathematically consistent.

    Checks the relationship: L ≈ N * dx

    Parameters:
    -----------
    rtol : float
        Relative tolerance for the floating point comparison.

    Returns:
    --------
    bool
        True if consistent, raises ValueError otherwise.
    """
    expected_L = self.N * self.dx
    if not jnp.isclose(self.L, expected_L, rtol=rtol):
        raise ValueError(
            f"Grid inconsistency detected.\n"
            f"  L (defined): {self.L}\n"
            f"  N * dx     : {expected_L}\n"
            f"  Difference : {abs(self.L - expected_L)}"
        )
    return True

from_N_L(N, L, dealias='2/3') classmethod

Initialize FourierGrid using Number of points (N) and Length (L).

Calculates: dx = L / N

Source code in spectraldiffx/_src/fourier/grid.py 
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@classmethod
def from_N_L(
    cls,
    N: int,
    L: float,
    dealias: Literal["2/3", "padding", None] | None = "2/3",
) -> "FourierGrid1D":
    """
    Initialize FourierGrid using Number of points (N) and Length (L).

    Calculates: dx = L / N
    """
    dx = L / N
    return cls(N=N, L=L, dx=dx, dealias=dealias)

from_N_dx(N, dx, dealias='2/3') classmethod

Initialize FourierGrid using Number of points (N) and Spacing (dx).

Calculates: L = N * dx

Source code in spectraldiffx/_src/fourier/grid.py 
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@classmethod
def from_N_dx(
    cls,
    N: int,
    dx: float,
    dealias: Literal["2/3", "padding", None] | None = "2/3",
) -> "FourierGrid1D":
    """
    Initialize FourierGrid using Number of points (N) and Spacing (dx).

    Calculates: L = N * dx
    """
    L = N * dx
    return cls(N=N, L=L, dx=dx, dealias=dealias)

from_L_dx(L, dx, dealias='2/3') classmethod

Initialize FourierGrid using Length (L) and Spacing (dx).

Calculates: N = L / dx (Must result in an integer)

Raises:

ValueError: If L is not divisible by dx (N is not an integer).

Source code in spectraldiffx/_src/fourier/grid.py 
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@classmethod
def from_L_dx(
    cls,
    L: float,
    dx: float,
    dealias: Literal["2/3", "padding", None] | None = "2/3",
) -> "FourierGrid1D":
    """
    Initialize FourierGrid using Length (L) and Spacing (dx).

    Calculates: N = L / dx (Must result in an integer)

    Raises:
    -------
    ValueError: If L is not divisible by dx (N is not an integer).
    """
    N_float = L / dx
    if not jnp.isclose(N_float % 1, 0) and not jnp.isclose(N_float % 1, 1):
        raise ValueError(
            f"L={L} is not divisible by dx={dx}. Resulting N={N_float} is not an integer."
        )

    N = int(round(N_float))
    return cls(N=N, L=L, dx=dx, dealias=dealias)

dealias_filter()

Dealiasing filter: 1 for kept modes, 0 for removed.

Returns:

filter : Array [N] Filter mask (1 or 0)

Source code in spectraldiffx/_src/fourier/grid.py 
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def dealias_filter(self) -> Float[Array, "N"]:
    """
    Dealiasing filter: 1 for kept modes, 0 for removed.

    Returns:
    --------
    filter : Array [N]
        Filter mask (1 or 0)
    """
    if self.dealias == "2/3":
        k = self.k
        k_max = jnp.abs(k).max()
        cutoff = k_max * 2 / 3
        return jnp.where(jnp.abs(k) <= cutoff, 1.0, 0.0)
    else:
        return jnp.ones(self.N)

transform(u, inverse=False)

Perform 1D Fourier Transform. Forward: physical -> spectral [u -> u_hat] Inverse: spectral -> physical [u_hat -> u]

Source code in spectraldiffx/_src/fourier/grid.py 
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def transform(self, u: Array, inverse: bool = False) -> Complex[Array, "N"]:
    """
    Perform 1D Fourier Transform.
    Forward: physical -> spectral [u -> u_hat]
    Inverse: spectral -> physical [u_hat -> u]
    """
    if inverse:
        return jnp.fft.ifft(u)
    return jnp.fft.fft(u)

FourierGrid2D

Bases: Module

2D Fourier grid for doubly periodic domain.

Mathematical Framework:

For domain [0, Lx] × [0, Ly]:

u(x,y) = ΣΣ û_{kx,ky}·exp(i(kx·x + ky·y))
2D FFT

û_{kx,ky} = (1/(Nx·Ny))·ΣΣ u_{j,l}·exp(-i(kx·x_j + ky·y_l))

Nx, Ny : int
    Grid points in x, y
Lx, Ly : float
    Domain lengths [m]
dx, dy : float
    Grid spacings in x, y [m]. Must satisfy L = N * d.
dealias : str
    Dealiasing method
Source code in spectraldiffx/_src/fourier/grid.py 
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class FourierGrid2D(eqx.Module):
    """
    2D Fourier grid for doubly periodic domain.

    Mathematical Framework:
    -----------------------
    For domain [0, Lx] × [0, Ly]:

        u(x,y) = ΣΣ û_{kx,ky}·exp(i(kx·x + ky·y))

    2D FFT:
        û_{kx,ky} = (1/(Nx·Ny))·ΣΣ u_{j,l}·exp(-i(kx·x_j + ky·y_l))

    Attributes:
    -----------
        Nx, Ny : int
            Grid points in x, y
        Lx, Ly : float
            Domain lengths [m]
        dx, dy : float
            Grid spacings in x, y [m]. Must satisfy L = N * d.
        dealias : str
            Dealiasing method
    """

    Nx: int
    Ny: int
    Lx: float
    Ly: float
    dx: float
    dy: float
    dealias: Literal["2/3", "padding", None] | None = "2/3"

    def check_consistency(self, rtol: float = 1e-5) -> bool:
        """
        Verifies that the grid attributes are mathematically consistent for both dimensions.

        Checks:
            Lx ≈ Nx * dx
            Ly ≈ Ny * dy

        Parameters:
        -----------
        rtol : float
            Relative tolerance for the floating point comparison.

        Returns:
        --------
        bool
            True if consistent, raises ValueError otherwise.
        """
        expected_Lx = self.Nx * self.dx
        expected_Ly = self.Ny * self.dy

        errors = []
        if not jnp.isclose(self.Lx, expected_Lx, rtol=rtol):
            errors.append(
                f"X-dimension inconsistency: Lx={self.Lx} vs Nx*dx={expected_Lx}"
            )
        if not jnp.isclose(self.Ly, expected_Ly, rtol=rtol):
            errors.append(
                f"Y-dimension inconsistency: Ly={self.Ly} vs Ny*dy={expected_Ly}"
            )

        if errors:
            raise ValueError("Grid inconsistency detected:\n" + "\n".join(errors))

        return True

    @classmethod
    def from_N_L(
        cls,
        Nx: int,
        Ny: int,
        Lx: float,
        Ly: float,
        dealias: Literal["2/3", "padding", None] | None = "2/3",
    ) -> "FourierGrid2D":
        """
        Initialize FourierGrid2D using Number of points (N) and Length (L).

        Calculates:
            dx = Lx / Nx
            dy = Ly / Ny
        """
        dx = Lx / Nx
        dy = Ly / Ny
        return cls(Nx=Nx, Ny=Ny, Lx=Lx, Ly=Ly, dx=dx, dy=dy, dealias=dealias)

    @classmethod
    def from_N_dx(
        cls,
        Nx: int,
        Ny: int,
        dx: float,
        dy: float,
        dealias: Literal["2/3", "padding", None] | None = "2/3",
    ) -> "FourierGrid2D":
        """
        Initialize FourierGrid2D using Number of points (N) and Spacing (dx/dy).

        Calculates:
            Lx = Nx * dx
            Ly = Ny * dy
        """
        Lx = Nx * dx
        Ly = Ny * dy
        return cls(Nx=Nx, Ny=Ny, Lx=Lx, Ly=Ly, dx=dx, dy=dy, dealias=dealias)

    @classmethod
    def from_L_dx(
        cls,
        Lx: float,
        Ly: float,
        dx: float,
        dy: float,
        dealias: Literal["2/3", "padding", None] | None = "2/3",
    ) -> "FourierGrid2D":
        """
        Initialize FourierGrid2D using Length (L) and Spacing (dx/dy).

        Calculates:
            Nx = Lx / dx
            Ny = Ly / dy

        Raises:
        -------
        ValueError: If L is not divisible by dx (N is not an integer).
        """
        Nx_float = Lx / dx
        Ny_float = Ly / dy

        errors = []
        if not jnp.isclose(Nx_float % 1, 0) and not jnp.isclose(Nx_float % 1, 1):
            errors.append(f"Lx={Lx} not divisible by dx={dx} (Nx={Nx_float})")
        if not jnp.isclose(Ny_float % 1, 0) and not jnp.isclose(Ny_float % 1, 1):
            errors.append(f"Ly={Ly} not divisible by dy={dy} (Ny={Ny_float})")

        if errors:
            raise ValueError("\n".join(errors))

        return cls(
            Nx=int(round(Nx_float)),
            Ny=int(round(Ny_float)),
            Lx=Lx,
            Ly=Ly,
            dx=dx,
            dy=dy,
            dealias=dealias,
        )

    @property
    def x(self) -> Float[Array, "Nx"]:
        """Physical grid points x: [0, dx, ..., Lx-dx]"""
        return jnp.linspace(0, self.Lx, self.Nx, endpoint=False)

    @property
    def y(self) -> Float[Array, "Ny"]:
        """Physical grid points y: [0, dy, ..., Ly-dy]"""
        return jnp.linspace(0, self.Ly, self.Ny, endpoint=False)

    @property
    def X(self) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
        """2D meshgrid: X, Y"""
        result = jnp.meshgrid(self.x, self.y, indexing="xy")
        return (result[0], result[1])

    @property
    def kx(self) -> Float[Array, "Nx"]:
        """Wavenumbers in x-direction"""
        return 2 * jnp.pi * jnp.fft.fftfreq(self.Nx, self.dx)

    @property
    def ky(self) -> Float[Array, "Ny"]:
        """Wavenumbers in y-direction"""
        return 2 * jnp.pi * jnp.fft.fftfreq(self.Ny, self.dy)

    @property
    def KX(self) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
        """2D wavenumber meshgrid: KX, KY"""
        result = jnp.meshgrid(self.kx, self.ky, indexing="xy")
        return (result[0], result[1])

    @property
    def K2(self) -> Float[Array, "Ny Nx"]:
        """
        Squared wavenumber magnitude: |k|² = kx² + ky²

        Used for Laplacian: ∇²u ↔ -|k|²·û
        """
        KX, KY = self.KX
        return KX**2 + KY**2

    def dealias_filter(self) -> Float[Array, "Ny Nx"]:
        """2D dealiasing filter."""
        if self.dealias == "2/3":
            KX, KY = self.KX
            kx_max = jnp.abs(self.kx).max()
            ky_max = jnp.abs(self.ky).max()
            cutoff_x = kx_max * 2 / 3
            cutoff_y = ky_max * 2 / 3

            filter_x = jnp.where(jnp.abs(KX) <= cutoff_x, 1.0, 0.0)
            filter_y = jnp.where(jnp.abs(KY) <= cutoff_y, 1.0, 0.0)

            return filter_x * filter_y
        else:
            return jnp.ones((self.Ny, self.Nx))

    def transform(self, u: Array, inverse: bool = False) -> Complex[Array, "Ny Nx"]:
        """Perform 2D Fourier Transform."""
        if inverse:
            return jnp.fft.ifft2(u)
        return jnp.fft.fft2(u)

Attributes

x property

Physical grid points x: [0, dx, ..., Lx-dx]

y property

Physical grid points y: [0, dy, ..., Ly-dy]

X property

2D meshgrid: X, Y

kx property

Wavenumbers in x-direction

ky property

Wavenumbers in y-direction

KX property

2D wavenumber meshgrid: KX, KY

K2 property

Squared wavenumber magnitude: |k|² = kx² + ky²

Used for Laplacian: ∇²u ↔ -|k|²·û

Functions

check_consistency(rtol=1e-05)

Verifies that the grid attributes are mathematically consistent for both dimensions.

Checks

Lx ≈ Nx * dx Ly ≈ Ny * dy

Parameters:

rtol : float Relative tolerance for the floating point comparison.

Returns:

bool True if consistent, raises ValueError otherwise.

Source code in spectraldiffx/_src/fourier/grid.py 
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def check_consistency(self, rtol: float = 1e-5) -> bool:
    """
    Verifies that the grid attributes are mathematically consistent for both dimensions.

    Checks:
        Lx ≈ Nx * dx
        Ly ≈ Ny * dy

    Parameters:
    -----------
    rtol : float
        Relative tolerance for the floating point comparison.

    Returns:
    --------
    bool
        True if consistent, raises ValueError otherwise.
    """
    expected_Lx = self.Nx * self.dx
    expected_Ly = self.Ny * self.dy

    errors = []
    if not jnp.isclose(self.Lx, expected_Lx, rtol=rtol):
        errors.append(
            f"X-dimension inconsistency: Lx={self.Lx} vs Nx*dx={expected_Lx}"
        )
    if not jnp.isclose(self.Ly, expected_Ly, rtol=rtol):
        errors.append(
            f"Y-dimension inconsistency: Ly={self.Ly} vs Ny*dy={expected_Ly}"
        )

    if errors:
        raise ValueError("Grid inconsistency detected:\n" + "\n".join(errors))

    return True

from_N_L(Nx, Ny, Lx, Ly, dealias='2/3') classmethod

Initialize FourierGrid2D using Number of points (N) and Length (L).

Calculates

dx = Lx / Nx dy = Ly / Ny

Source code in spectraldiffx/_src/fourier/grid.py 
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@classmethod
def from_N_L(
    cls,
    Nx: int,
    Ny: int,
    Lx: float,
    Ly: float,
    dealias: Literal["2/3", "padding", None] | None = "2/3",
) -> "FourierGrid2D":
    """
    Initialize FourierGrid2D using Number of points (N) and Length (L).

    Calculates:
        dx = Lx / Nx
        dy = Ly / Ny
    """
    dx = Lx / Nx
    dy = Ly / Ny
    return cls(Nx=Nx, Ny=Ny, Lx=Lx, Ly=Ly, dx=dx, dy=dy, dealias=dealias)

from_N_dx(Nx, Ny, dx, dy, dealias='2/3') classmethod

Initialize FourierGrid2D using Number of points (N) and Spacing (dx/dy).

Calculates

Lx = Nx * dx Ly = Ny * dy

Source code in spectraldiffx/_src/fourier/grid.py 
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@classmethod
def from_N_dx(
    cls,
    Nx: int,
    Ny: int,
    dx: float,
    dy: float,
    dealias: Literal["2/3", "padding", None] | None = "2/3",
) -> "FourierGrid2D":
    """
    Initialize FourierGrid2D using Number of points (N) and Spacing (dx/dy).

    Calculates:
        Lx = Nx * dx
        Ly = Ny * dy
    """
    Lx = Nx * dx
    Ly = Ny * dy
    return cls(Nx=Nx, Ny=Ny, Lx=Lx, Ly=Ly, dx=dx, dy=dy, dealias=dealias)

from_L_dx(Lx, Ly, dx, dy, dealias='2/3') classmethod

Initialize FourierGrid2D using Length (L) and Spacing (dx/dy).

Calculates

Nx = Lx / dx Ny = Ly / dy

Raises:

ValueError: If L is not divisible by dx (N is not an integer).

Source code in spectraldiffx/_src/fourier/grid.py 
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@classmethod
def from_L_dx(
    cls,
    Lx: float,
    Ly: float,
    dx: float,
    dy: float,
    dealias: Literal["2/3", "padding", None] | None = "2/3",
) -> "FourierGrid2D":
    """
    Initialize FourierGrid2D using Length (L) and Spacing (dx/dy).

    Calculates:
        Nx = Lx / dx
        Ny = Ly / dy

    Raises:
    -------
    ValueError: If L is not divisible by dx (N is not an integer).
    """
    Nx_float = Lx / dx
    Ny_float = Ly / dy

    errors = []
    if not jnp.isclose(Nx_float % 1, 0) and not jnp.isclose(Nx_float % 1, 1):
        errors.append(f"Lx={Lx} not divisible by dx={dx} (Nx={Nx_float})")
    if not jnp.isclose(Ny_float % 1, 0) and not jnp.isclose(Ny_float % 1, 1):
        errors.append(f"Ly={Ly} not divisible by dy={dy} (Ny={Ny_float})")

    if errors:
        raise ValueError("\n".join(errors))

    return cls(
        Nx=int(round(Nx_float)),
        Ny=int(round(Ny_float)),
        Lx=Lx,
        Ly=Ly,
        dx=dx,
        dy=dy,
        dealias=dealias,
    )

dealias_filter()

2D dealiasing filter.

Source code in spectraldiffx/_src/fourier/grid.py 
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def dealias_filter(self) -> Float[Array, "Ny Nx"]:
    """2D dealiasing filter."""
    if self.dealias == "2/3":
        KX, KY = self.KX
        kx_max = jnp.abs(self.kx).max()
        ky_max = jnp.abs(self.ky).max()
        cutoff_x = kx_max * 2 / 3
        cutoff_y = ky_max * 2 / 3

        filter_x = jnp.where(jnp.abs(KX) <= cutoff_x, 1.0, 0.0)
        filter_y = jnp.where(jnp.abs(KY) <= cutoff_y, 1.0, 0.0)

        return filter_x * filter_y
    else:
        return jnp.ones((self.Ny, self.Nx))

transform(u, inverse=False)

Perform 2D Fourier Transform.

Source code in spectraldiffx/_src/fourier/grid.py 
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def transform(self, u: Array, inverse: bool = False) -> Complex[Array, "Ny Nx"]:
    """Perform 2D Fourier Transform."""
    if inverse:
        return jnp.fft.ifft2(u)
    return jnp.fft.fft2(u)

FourierGrid3D

Bases: Module

3D Fourier grid for triply periodic domain.

Mathematical Framework:

For domain [0, Lz] × [0, Ly] × [0, Lx]:

u(z,y,x) = ΣΣΣ û_{kz,ky,kx}·exp(i(kz·z + ky·y + kx·x))
3D FFT

û_{kz,ky,kx} = (1/(Nz·Ny·Nx))·ΣΣΣ u_{m,l,j}·exp(-i(kz·z_m + ky·y_l + kx·x_j))

Grid Shapes (indexing='ij'): Physical/Spectral arrays have shape (Nz, Ny, Nx). Axis 0: z-direction (Depth) Axis 1: y-direction (Height) Axis 2: x-direction (Width)

Nz, Ny, Nx : int
    Grid points in z, y, x
Lz, Ly, Lx : float
    Domain lengths [m]
dz, dy, dx : float
    Grid spacings in z, y, x [m]. Must satisfy L = N * d.
dealias : str
    Dealiasing method
Source code in spectraldiffx/_src/fourier/grid.py 
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class FourierGrid3D(eqx.Module):
    """
    3D Fourier grid for triply periodic domain.

    Mathematical Framework:
    -----------------------
    For domain [0, Lz] × [0, Ly] × [0, Lx]:

        u(z,y,x) = ΣΣΣ û_{kz,ky,kx}·exp(i(kz·z + ky·y + kx·x))

    3D FFT:
        û_{kz,ky,kx} = (1/(Nz·Ny·Nx))·ΣΣΣ u_{m,l,j}·exp(-i(kz·z_m + ky·y_l + kx·x_j))

    Grid Shapes (indexing='ij'):
        Physical/Spectral arrays have shape (Nz, Ny, Nx).
        Axis 0: z-direction (Depth)
        Axis 1: y-direction (Height)
        Axis 2: x-direction (Width)

    Attributes:
    -----------
        Nz, Ny, Nx : int
            Grid points in z, y, x
        Lz, Ly, Lx : float
            Domain lengths [m]
        dz, dy, dx : float
            Grid spacings in z, y, x [m]. Must satisfy L = N * d.
        dealias : str
            Dealiasing method
    """

    Nz: int
    Ny: int
    Nx: int
    Lz: float
    Ly: float
    Lx: float
    dz: float
    dy: float
    dx: float
    dealias: Literal["2/3", "padding", None] | None = "2/3"

    def check_consistency(self, rtol: float = 1e-5) -> bool:
        """
        Verifies that the grid attributes are mathematically consistent for all three dimensions.

        Checks:
            Lz ≈ Nz * dz
            Ly ≈ Ny * dy
            Lx ≈ Nx * dx

        Parameters:
        -----------
        rtol : float
            Relative tolerance for the floating point comparison.

        Returns:
        --------
        bool
            True if consistent, raises ValueError otherwise.
        """
        expected_Lz = self.Nz * self.dz
        expected_Ly = self.Ny * self.dy
        expected_Lx = self.Nx * self.dx

        errors = []
        if not jnp.isclose(self.Lz, expected_Lz, rtol=rtol):
            errors.append(
                f"Z-dimension inconsistency: Lz={self.Lz} vs Nz*dz={expected_Lz}"
            )
        if not jnp.isclose(self.Ly, expected_Ly, rtol=rtol):
            errors.append(
                f"Y-dimension inconsistency: Ly={self.Ly} vs Ny*dy={expected_Ly}"
            )
        if not jnp.isclose(self.Lx, expected_Lx, rtol=rtol):
            errors.append(
                f"X-dimension inconsistency: Lx={self.Lx} vs Nx*dx={expected_Lx}"
            )

        if errors:
            raise ValueError("Grid inconsistency detected:\n" + "\n".join(errors))

        return True

    @classmethod
    def from_N_L(
        cls,
        Nz: int,
        Ny: int,
        Nx: int,
        Lz: float,
        Ly: float,
        Lx: float,
        dealias: Literal["2/3", "padding", None] | None = "2/3",
    ) -> "FourierGrid3D":
        """
        Initialize FourierGrid3D using Number of points (N) and Length (L).

        Calculates:
            dz = Lz / Nz
            dy = Ly / Ny
            dx = Lx / Nx
        """
        dz = Lz / Nz
        dy = Ly / Ny
        dx = Lx / Nx
        return cls(
            Nz=Nz,
            Ny=Ny,
            Nx=Nx,
            Lz=Lz,
            Ly=Ly,
            Lx=Lx,
            dz=dz,
            dy=dy,
            dx=dx,
            dealias=dealias,
        )

    @classmethod
    def from_N_dx(
        cls,
        Nz: int,
        Ny: int,
        Nx: int,
        dz: float,
        dy: float,
        dx: float,
        dealias: Literal["2/3", "padding", None] | None = "2/3",
    ) -> "FourierGrid3D":
        """
        Initialize FourierGrid3D using Number of points (N) and Spacing (dz/dy/dx).

        Calculates:
            Lz = Nz * dz
            Ly = Ny * dy
            Lx = Nx * dx
        """
        Lz = Nz * dz
        Ly = Ny * dy
        Lx = Nx * dx
        return cls(
            Nz=Nz,
            Ny=Ny,
            Nx=Nx,
            Lz=Lz,
            Ly=Ly,
            Lx=Lx,
            dz=dz,
            dy=dy,
            dx=dx,
            dealias=dealias,
        )

    @classmethod
    def from_L_dx(
        cls,
        Lz: float,
        Ly: float,
        Lx: float,
        dz: float,
        dy: float,
        dx: float,
        dealias: Literal["2/3", "padding", None] | None = "2/3",
    ) -> "FourierGrid3D":
        """
        Initialize FourierGrid3D using Length (L) and Spacing (dz/dy/dx).

        Calculates:
            Nz = Lz / dz
            Ny = Ly / dy
            Nx = Lx / dx

        Raises:
        -------
        ValueError: If any L is not divisible by its corresponding d.
        """
        Nz_float = Lz / dz
        Ny_float = Ly / dy
        Nx_float = Lx / dx

        errors = []
        if not jnp.isclose(Nz_float % 1, 0) and not jnp.isclose(Nz_float % 1, 1):
            errors.append(f"Lz={Lz} not divisible by dz={dz} (Nz={Nz_float})")
        if not jnp.isclose(Ny_float % 1, 0) and not jnp.isclose(Ny_float % 1, 1):
            errors.append(f"Ly={Ly} not divisible by dy={dy} (Ny={Ny_float})")
        if not jnp.isclose(Nx_float % 1, 0) and not jnp.isclose(Nx_float % 1, 1):
            errors.append(f"Lx={Lx} not divisible by dx={dx} (Nx={Nx_float})")

        if errors:
            raise ValueError("\n".join(errors))

        return cls(
            Nz=int(round(Nz_float)),
            Ny=int(round(Ny_float)),
            Nx=int(round(Nx_float)),
            Lz=Lz,
            Ly=Ly,
            Lx=Lx,
            dz=dz,
            dy=dy,
            dx=dx,
            dealias=dealias,
        )

    @property
    def z(self) -> Float[Array, "Nz"]:
        """Physical grid points z: [0, dz, ..., Lz-dz]"""
        return jnp.linspace(0, self.Lz, self.Nz, endpoint=False)

    @property
    def y(self) -> Float[Array, "Ny"]:
        """Physical grid points y: [0, dy, ..., Ly-dy]"""
        return jnp.linspace(0, self.Ly, self.Ny, endpoint=False)

    @property
    def x(self) -> Float[Array, "Nx"]:
        """Physical grid points x: [0, dx, ..., Lx-dx]"""
        return jnp.linspace(0, self.Lx, self.Nx, endpoint=False)

    @property
    def X(
        self,
    ) -> tuple[
        Float[Array, "Nz Ny Nx"], Float[Array, "Nz Ny Nx"], Float[Array, "Nz Ny Nx"]
    ]:
        """
        3D meshgrid: Z, Y, X

        Shape Note:
        -----------
        Uses indexing='ij' with (z, y, x) input.
        Resulting shape: (Nz, Ny, Nx).
        """
        result = jnp.meshgrid(self.z, self.y, self.x, indexing="ij")
        return (result[0], result[1], result[2])

    @property
    def kz(self) -> Float[Array, "Nz"]:
        """Wavenumbers in z-direction"""
        return 2 * jnp.pi * jnp.fft.fftfreq(self.Nz, self.dz)

    @property
    def ky(self) -> Float[Array, "Ny"]:
        """Wavenumbers in y-direction"""
        return 2 * jnp.pi * jnp.fft.fftfreq(self.Ny, self.dy)

    @property
    def kx(self) -> Float[Array, "Nx"]:
        """Wavenumbers in x-direction"""
        return 2 * jnp.pi * jnp.fft.fftfreq(self.Nx, self.dx)

    @property
    def KX(
        self,
    ) -> tuple[
        Float[Array, "Nz Ny Nx"], Float[Array, "Nz Ny Nx"], Float[Array, "Nz Ny Nx"]
    ]:
        """
        3D wavenumber meshgrid: KZ, KY, KX

        Shape Note:
        -----------
        Uses indexing='ij' with (kz, ky, kx) input.
        Resulting shape: (Nz, Ny, Nx).
        """
        result = jnp.meshgrid(self.kz, self.ky, self.kx, indexing="ij")
        return (result[0], result[1], result[2])

    @property
    def K2(self) -> Float[Array, "Nz Ny Nx"]:
        """
        Squared wavenumber magnitude: |k|² = kz² + ky² + kx²

        Used for Laplacian: ∇²u ↔ -|k|²·û
        """
        KZ, KY, KX = self.KX
        return KZ**2 + KY**2 + KX**2

    def dealias_filter(self) -> Float[Array, "Nz Ny Nx"]:
        """3D dealiasing filter."""
        if self.dealias == "2/3":
            KZ, KY, KX = self.KX
            kz_max = jnp.abs(self.kz).max()
            ky_max = jnp.abs(self.ky).max()
            kx_max = jnp.abs(self.kx).max()

            cutoff_z = kz_max * 2 / 3
            cutoff_y = ky_max * 2 / 3
            cutoff_x = kx_max * 2 / 3

            filter_z = jnp.where(jnp.abs(KZ) <= cutoff_z, 1.0, 0.0)
            filter_y = jnp.where(jnp.abs(KY) <= cutoff_y, 1.0, 0.0)
            filter_x = jnp.where(jnp.abs(KX) <= cutoff_x, 1.0, 0.0)

            return filter_z * filter_y * filter_x
        else:
            return jnp.ones((self.Nz, self.Ny, self.Nx))

    def transform(self, u: Array, inverse: bool = False) -> Complex[Array, "Nz Ny Nx"]:
        """Perform 3D Fourier Transform (Nz, Ny, Nx)."""
        if inverse:
            return jnp.fft.ifftn(u)
        return jnp.fft.fftn(u)

Attributes

z property

Physical grid points z: [0, dz, ..., Lz-dz]

y property

Physical grid points y: [0, dy, ..., Ly-dy]

x property

Physical grid points x: [0, dx, ..., Lx-dx]

X property

3D meshgrid: Z, Y, X

Shape Note:

Uses indexing='ij' with (z, y, x) input. Resulting shape: (Nz, Ny, Nx).

kz property

Wavenumbers in z-direction

ky property

Wavenumbers in y-direction

kx property

Wavenumbers in x-direction

KX property

3D wavenumber meshgrid: KZ, KY, KX

Shape Note:

Uses indexing='ij' with (kz, ky, kx) input. Resulting shape: (Nz, Ny, Nx).

K2 property

Squared wavenumber magnitude: |k|² = kz² + ky² + kx²

Used for Laplacian: ∇²u ↔ -|k|²·û

Functions

check_consistency(rtol=1e-05)

Verifies that the grid attributes are mathematically consistent for all three dimensions.

Checks

Lz ≈ Nz * dz Ly ≈ Ny * dy Lx ≈ Nx * dx

Parameters:

rtol : float Relative tolerance for the floating point comparison.

Returns:

bool True if consistent, raises ValueError otherwise.

Source code in spectraldiffx/_src/fourier/grid.py 
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def check_consistency(self, rtol: float = 1e-5) -> bool:
    """
    Verifies that the grid attributes are mathematically consistent for all three dimensions.

    Checks:
        Lz ≈ Nz * dz
        Ly ≈ Ny * dy
        Lx ≈ Nx * dx

    Parameters:
    -----------
    rtol : float
        Relative tolerance for the floating point comparison.

    Returns:
    --------
    bool
        True if consistent, raises ValueError otherwise.
    """
    expected_Lz = self.Nz * self.dz
    expected_Ly = self.Ny * self.dy
    expected_Lx = self.Nx * self.dx

    errors = []
    if not jnp.isclose(self.Lz, expected_Lz, rtol=rtol):
        errors.append(
            f"Z-dimension inconsistency: Lz={self.Lz} vs Nz*dz={expected_Lz}"
        )
    if not jnp.isclose(self.Ly, expected_Ly, rtol=rtol):
        errors.append(
            f"Y-dimension inconsistency: Ly={self.Ly} vs Ny*dy={expected_Ly}"
        )
    if not jnp.isclose(self.Lx, expected_Lx, rtol=rtol):
        errors.append(
            f"X-dimension inconsistency: Lx={self.Lx} vs Nx*dx={expected_Lx}"
        )

    if errors:
        raise ValueError("Grid inconsistency detected:\n" + "\n".join(errors))

    return True

from_N_L(Nz, Ny, Nx, Lz, Ly, Lx, dealias='2/3') classmethod

Initialize FourierGrid3D using Number of points (N) and Length (L).

Calculates

dz = Lz / Nz dy = Ly / Ny dx = Lx / Nx

Source code in spectraldiffx/_src/fourier/grid.py 
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@classmethod
def from_N_L(
    cls,
    Nz: int,
    Ny: int,
    Nx: int,
    Lz: float,
    Ly: float,
    Lx: float,
    dealias: Literal["2/3", "padding", None] | None = "2/3",
) -> "FourierGrid3D":
    """
    Initialize FourierGrid3D using Number of points (N) and Length (L).

    Calculates:
        dz = Lz / Nz
        dy = Ly / Ny
        dx = Lx / Nx
    """
    dz = Lz / Nz
    dy = Ly / Ny
    dx = Lx / Nx
    return cls(
        Nz=Nz,
        Ny=Ny,
        Nx=Nx,
        Lz=Lz,
        Ly=Ly,
        Lx=Lx,
        dz=dz,
        dy=dy,
        dx=dx,
        dealias=dealias,
    )

from_N_dx(Nz, Ny, Nx, dz, dy, dx, dealias='2/3') classmethod

Initialize FourierGrid3D using Number of points (N) and Spacing (dz/dy/dx).

Calculates

Lz = Nz * dz Ly = Ny * dy Lx = Nx * dx

Source code in spectraldiffx/_src/fourier/grid.py 
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@classmethod
def from_N_dx(
    cls,
    Nz: int,
    Ny: int,
    Nx: int,
    dz: float,
    dy: float,
    dx: float,
    dealias: Literal["2/3", "padding", None] | None = "2/3",
) -> "FourierGrid3D":
    """
    Initialize FourierGrid3D using Number of points (N) and Spacing (dz/dy/dx).

    Calculates:
        Lz = Nz * dz
        Ly = Ny * dy
        Lx = Nx * dx
    """
    Lz = Nz * dz
    Ly = Ny * dy
    Lx = Nx * dx
    return cls(
        Nz=Nz,
        Ny=Ny,
        Nx=Nx,
        Lz=Lz,
        Ly=Ly,
        Lx=Lx,
        dz=dz,
        dy=dy,
        dx=dx,
        dealias=dealias,
    )

from_L_dx(Lz, Ly, Lx, dz, dy, dx, dealias='2/3') classmethod

Initialize FourierGrid3D using Length (L) and Spacing (dz/dy/dx).

Calculates

Nz = Lz / dz Ny = Ly / dy Nx = Lx / dx

Raises:

ValueError: If any L is not divisible by its corresponding d.

Source code in spectraldiffx/_src/fourier/grid.py 
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@classmethod
def from_L_dx(
    cls,
    Lz: float,
    Ly: float,
    Lx: float,
    dz: float,
    dy: float,
    dx: float,
    dealias: Literal["2/3", "padding", None] | None = "2/3",
) -> "FourierGrid3D":
    """
    Initialize FourierGrid3D using Length (L) and Spacing (dz/dy/dx).

    Calculates:
        Nz = Lz / dz
        Ny = Ly / dy
        Nx = Lx / dx

    Raises:
    -------
    ValueError: If any L is not divisible by its corresponding d.
    """
    Nz_float = Lz / dz
    Ny_float = Ly / dy
    Nx_float = Lx / dx

    errors = []
    if not jnp.isclose(Nz_float % 1, 0) and not jnp.isclose(Nz_float % 1, 1):
        errors.append(f"Lz={Lz} not divisible by dz={dz} (Nz={Nz_float})")
    if not jnp.isclose(Ny_float % 1, 0) and not jnp.isclose(Ny_float % 1, 1):
        errors.append(f"Ly={Ly} not divisible by dy={dy} (Ny={Ny_float})")
    if not jnp.isclose(Nx_float % 1, 0) and not jnp.isclose(Nx_float % 1, 1):
        errors.append(f"Lx={Lx} not divisible by dx={dx} (Nx={Nx_float})")

    if errors:
        raise ValueError("\n".join(errors))

    return cls(
        Nz=int(round(Nz_float)),
        Ny=int(round(Ny_float)),
        Nx=int(round(Nx_float)),
        Lz=Lz,
        Ly=Ly,
        Lx=Lx,
        dz=dz,
        dy=dy,
        dx=dx,
        dealias=dealias,
    )

dealias_filter()

3D dealiasing filter.

Source code in spectraldiffx/_src/fourier/grid.py 
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def dealias_filter(self) -> Float[Array, "Nz Ny Nx"]:
    """3D dealiasing filter."""
    if self.dealias == "2/3":
        KZ, KY, KX = self.KX
        kz_max = jnp.abs(self.kz).max()
        ky_max = jnp.abs(self.ky).max()
        kx_max = jnp.abs(self.kx).max()

        cutoff_z = kz_max * 2 / 3
        cutoff_y = ky_max * 2 / 3
        cutoff_x = kx_max * 2 / 3

        filter_z = jnp.where(jnp.abs(KZ) <= cutoff_z, 1.0, 0.0)
        filter_y = jnp.where(jnp.abs(KY) <= cutoff_y, 1.0, 0.0)
        filter_x = jnp.where(jnp.abs(KX) <= cutoff_x, 1.0, 0.0)

        return filter_z * filter_y * filter_x
    else:
        return jnp.ones((self.Nz, self.Ny, self.Nx))

transform(u, inverse=False)

Perform 3D Fourier Transform (Nz, Ny, Nx).

Source code in spectraldiffx/_src/fourier/grid.py 
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def transform(self, u: Array, inverse: bool = False) -> Complex[Array, "Nz Ny Nx"]:
    """Perform 3D Fourier Transform (Nz, Ny, Nx)."""
    if inverse:
        return jnp.fft.ifftn(u)
    return jnp.fft.fftn(u)