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Spherical Harmonic Solvers

SphericalPoissonSolver

Bases: Module

Spectral Poisson solver on the sphere.

Solves: nabla^2 phi = f

In spectral space (SHT coefficients): phi_hat(l, m) = -f_hat(l, m) / [l*(l+1)/R^2]

The l=0 mode is set to zero (mean of phi is undetermined for Poisson).

Attributes:

grid : SphericalGrid2D or SphericalGrid1D The spherical grid.

Source code in spectraldiffx/_src/spherical/solvers.py 
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class SphericalPoissonSolver(eqx.Module):
    """
    Spectral Poisson solver on the sphere.

    Solves: nabla^2 phi = f

    In spectral space (SHT coefficients):
        phi_hat(l, m) = -f_hat(l, m) / [l*(l+1)/R^2]

    The l=0 mode is set to zero (mean of phi is undetermined for Poisson).

    Attributes:
    -----------
    grid : SphericalGrid2D or SphericalGrid1D
        The spherical grid.
    """

    grid: object  # SphericalGrid2D or SphericalGrid1D

    def solve(
        self,
        f: Float[Array, "..."],
        zero_mean: bool = True,
        spectral: bool = False,
    ) -> Float[Array, "..."]:
        """
        Solve nabla^2 phi = f on the sphere.

        Parameters:
        -----------
        f : Array
            Source field (physical space or spectral if spectral=True).
            Shape (Ny,) for 1D or (Ny, Nx) for 2D.
        zero_mean : bool
            If True, set the l=0 (global mean) mode to zero.
        spectral : bool
            If True, treat f as SHT/DLT coefficients.

        Returns:
        --------
        phi : Array
            Solution field in physical space.
        """
        R = (
            self.grid.L / jnp.pi
            if isinstance(self.grid, SphericalGrid1D)
            else self.grid.Ly / jnp.pi
        )
        f_hat = f if spectral else self.grid.transform(f)
        l = self.grid.l  # (N,) or (Ny,)
        eigenval = l * (l + 1) / (R**2)

        if isinstance(self.grid, SphericalGrid1D):
            # 1D case
            denom = jnp.where(eigenval == 0.0, 1.0, eigenval)
            phi_hat = -f_hat / denom
            if zero_mean:
                phi_hat = jnp.where(l == 0.0, 0.0, phi_hat)
        else:
            # 2D case: eigenval is (Ny,), broadcast to (Ny, Nx)
            denom = jnp.where(eigenval[:, None] == 0.0, 1.0, eigenval[:, None])
            phi_hat = -f_hat / denom
            if zero_mean:
                phi_hat = jnp.where(l[:, None] == 0.0, 0.0, phi_hat)

        return self.grid.transform(phi_hat, inverse=True)

Functions

solve(f, zero_mean=True, spectral=False)

Solve nabla^2 phi = f on the sphere.

Parameters:

f : Array Source field (physical space or spectral if spectral=True). Shape (Ny,) for 1D or (Ny, Nx) for 2D. zero_mean : bool If True, set the l=0 (global mean) mode to zero. spectral : bool If True, treat f as SHT/DLT coefficients.

Returns:

phi : Array Solution field in physical space.

Source code in spectraldiffx/_src/spherical/solvers.py 
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def solve(
    self,
    f: Float[Array, "..."],
    zero_mean: bool = True,
    spectral: bool = False,
) -> Float[Array, "..."]:
    """
    Solve nabla^2 phi = f on the sphere.

    Parameters:
    -----------
    f : Array
        Source field (physical space or spectral if spectral=True).
        Shape (Ny,) for 1D or (Ny, Nx) for 2D.
    zero_mean : bool
        If True, set the l=0 (global mean) mode to zero.
    spectral : bool
        If True, treat f as SHT/DLT coefficients.

    Returns:
    --------
    phi : Array
        Solution field in physical space.
    """
    R = (
        self.grid.L / jnp.pi
        if isinstance(self.grid, SphericalGrid1D)
        else self.grid.Ly / jnp.pi
    )
    f_hat = f if spectral else self.grid.transform(f)
    l = self.grid.l  # (N,) or (Ny,)
    eigenval = l * (l + 1) / (R**2)

    if isinstance(self.grid, SphericalGrid1D):
        # 1D case
        denom = jnp.where(eigenval == 0.0, 1.0, eigenval)
        phi_hat = -f_hat / denom
        if zero_mean:
            phi_hat = jnp.where(l == 0.0, 0.0, phi_hat)
    else:
        # 2D case: eigenval is (Ny,), broadcast to (Ny, Nx)
        denom = jnp.where(eigenval[:, None] == 0.0, 1.0, eigenval[:, None])
        phi_hat = -f_hat / denom
        if zero_mean:
            phi_hat = jnp.where(l[:, None] == 0.0, 0.0, phi_hat)

    return self.grid.transform(phi_hat, inverse=True)

SphericalHelmholtzSolver

Bases: Module

Spectral Helmholtz solver on the sphere.

Solves: (nabla^2 - alpha) phi = f

In spectral space

phi_hat(l, m) = -f_hat(l, m) / [l*(l+1)/R^2 + alpha]

Attributes:

grid : SphericalGrid2D or SphericalGrid1D The spherical grid.

Source code in spectraldiffx/_src/spherical/solvers.py 
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class SphericalHelmholtzSolver(eqx.Module):
    """
    Spectral Helmholtz solver on the sphere.

    Solves: (nabla^2 - alpha) phi = f

    In spectral space:
        phi_hat(l, m) = -f_hat(l, m) / [l*(l+1)/R^2 + alpha]

    Attributes:
    -----------
    grid : SphericalGrid2D or SphericalGrid1D
        The spherical grid.
    """

    grid: object  # SphericalGrid2D or SphericalGrid1D

    def solve(
        self,
        f: Float[Array, "..."],
        alpha: float = 0.0,
        zero_mean: bool = True,
        spectral: bool = False,
    ) -> Float[Array, "..."]:
        """
        Solve (nabla^2 - alpha) phi = f on the sphere.

        Parameters:
        -----------
        f : Array
            Source field (physical or spectral).
        alpha : float
            Helmholtz parameter (>= 0).  alpha=0 reduces to Poisson.
        zero_mean : bool
            If True, force l=0 (mean) mode to zero.  Relevant when alpha=0.
        spectral : bool
            If True, treat f as SHT/DLT coefficients.

        Returns:
        --------
        phi : Array
            Solution field in physical space.
        """
        R = (
            self.grid.L / jnp.pi
            if isinstance(self.grid, SphericalGrid1D)
            else self.grid.Ly / jnp.pi
        )
        f_hat = f if spectral else self.grid.transform(f)
        l = self.grid.l
        eigenval = l * (l + 1) / (R**2)

        if isinstance(self.grid, SphericalGrid1D):
            denom = eigenval + alpha
            denom_safe = jnp.where(denom == 0.0, 1.0, denom)
            phi_hat = -f_hat / denom_safe
            if zero_mean:
                phi_hat = jnp.where(l == 0.0, 0.0, phi_hat)
        else:
            denom = eigenval[:, None] + alpha
            denom_safe = jnp.where(denom == 0.0, 1.0, denom)
            phi_hat = -f_hat / denom_safe
            if zero_mean:
                phi_hat = jnp.where(l[:, None] == 0.0, 0.0, phi_hat)

        return self.grid.transform(phi_hat, inverse=True)

Functions

solve(f, alpha=0.0, zero_mean=True, spectral=False)

Solve (nabla^2 - alpha) phi = f on the sphere.

Parameters:

f : Array Source field (physical or spectral). alpha : float Helmholtz parameter (>= 0). alpha=0 reduces to Poisson. zero_mean : bool If True, force l=0 (mean) mode to zero. Relevant when alpha=0. spectral : bool If True, treat f as SHT/DLT coefficients.

Returns:

phi : Array Solution field in physical space.

Source code in spectraldiffx/_src/spherical/solvers.py 
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def solve(
    self,
    f: Float[Array, "..."],
    alpha: float = 0.0,
    zero_mean: bool = True,
    spectral: bool = False,
) -> Float[Array, "..."]:
    """
    Solve (nabla^2 - alpha) phi = f on the sphere.

    Parameters:
    -----------
    f : Array
        Source field (physical or spectral).
    alpha : float
        Helmholtz parameter (>= 0).  alpha=0 reduces to Poisson.
    zero_mean : bool
        If True, force l=0 (mean) mode to zero.  Relevant when alpha=0.
    spectral : bool
        If True, treat f as SHT/DLT coefficients.

    Returns:
    --------
    phi : Array
        Solution field in physical space.
    """
    R = (
        self.grid.L / jnp.pi
        if isinstance(self.grid, SphericalGrid1D)
        else self.grid.Ly / jnp.pi
    )
    f_hat = f if spectral else self.grid.transform(f)
    l = self.grid.l
    eigenval = l * (l + 1) / (R**2)

    if isinstance(self.grid, SphericalGrid1D):
        denom = eigenval + alpha
        denom_safe = jnp.where(denom == 0.0, 1.0, denom)
        phi_hat = -f_hat / denom_safe
        if zero_mean:
            phi_hat = jnp.where(l == 0.0, 0.0, phi_hat)
    else:
        denom = eigenval[:, None] + alpha
        denom_safe = jnp.where(denom == 0.0, 1.0, denom)
        phi_hat = -f_hat / denom_safe
        if zero_mean:
            phi_hat = jnp.where(l[:, None] == 0.0, 0.0, phi_hat)

    return self.grid.transform(phi_hat, inverse=True)