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Spectral Derivative Operators

SpectralDerivative1D

Bases: Module

1D Spectral derivative operator using the Fast Fourier Transform (FFT).

This class provides methods to differentiate fields periodically using spectral accuracy. It leverages the property that the Fourier Transform (FT) turns differentiation into multiplication by the wavenumber vector.

Mathematical Formulation:

Given a periodic field u(x) on a domain of length L with N points: u(x) = Σ u_hat(k) * exp(i * k * x)

The n-th derivative is

d^n u / dx^n = Σ (i * k)^n * u_hat(k) * exp(i * k * x)

where k are the discrete wavenumbers: k = 2 * pi * n / L.

grid : FourierGrid1D
    The 1D grid object containing wavenumbers k [N].
Source code in spectraldiffx/_src/fourier/operators.py 
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class SpectralDerivative1D(eqx.Module):
    """
    1D Spectral derivative operator using the Fast Fourier Transform (FFT).

    This class provides methods to differentiate fields periodically using
    spectral accuracy. It leverages the property that the Fourier Transform (FT)
    turns differentiation into multiplication by the wavenumber vector.

    Mathematical Formulation:
    -------------------------
    Given a periodic field u(x) on a domain of length L with N points:
        u(x) = Σ u_hat(k) * exp(i * k * x)

    The n-th derivative is:
        d^n u / dx^n = Σ (i * k)^n * u_hat(k) * exp(i * k * x)

    where k are the discrete wavenumbers: k = 2 * pi * n / L.

    Attributes:
    -----------
        grid : FourierGrid1D
            The 1D grid object containing wavenumbers k [N].
    """

    grid: FourierGrid1D

    def __call__(
        self, u: Array, order: int = 1, spectral: bool = False
    ) -> Float[Array, "N"]:
        """
        Compute the n-th derivative of a field.

        Parameters:
        -----------
        u : Array [N]
            The input field. If spectral=False, this is physical space.
            If spectral=True, this is complex Fourier coefficients.
        order : int, optional
            The order of the derivative (1=first, 2=second, etc.). Default is 1.
        spectral : bool, optional
            Whether the input 'u' is already in Fourier space. Default is False.

        Returns:
        --------
        du_dx : Array [N]
            The n-th derivative in physical space.
        """
        # 1. Obtain spectral coefficients u_hat [N]
        u_hat = u if spectral else self.grid.transform(u)

        # 2. Multiply by (i*k)^order [N]
        # We use k_dealias to zero out high-frequency modes prone to aliasing
        k = self.grid.k_dealias
        du_hat = (1j * k) ** order * u_hat

        # 3. Inverse Transform back to physical space [N]
        return self.grid.transform(du_hat, inverse=True).real

    def gradient(self, u: Array, spectral: bool = False) -> Array:
        """
        Compute the first derivative (gradient) du/dx.

        Parameters:
        -----------
        u : Array [N]
            Input field.
        spectral : bool, optional
            If True, treats u as Fourier coefficients.
        """
        return self(u, order=1, spectral=spectral)

    def laplacian(self, u: Array, spectral: bool = False) -> Array:
        """
        Compute the second derivative (Laplacian) d^2u/dx^2.

        Operation in Fourier Space:
            lap_hat = -(k^2) * u_hat
        """
        u_hat = u if spectral else self.grid.transform(u)
        k = self.grid.k
        dealias = self.grid.dealias_filter()
        lap_hat = -(k**2) * u_hat * dealias
        return self.grid.transform(lap_hat, inverse=True).real

    def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "N"]:
        """Compute the biharmonic operator: d^4u/dx^4.

        In Fourier space: k^4 * u_hat.
        """
        u_hat = u if spectral else self.grid.transform(u)
        k = self.grid.k
        dealias = self.grid.dealias_filter()
        bih_hat = k**4 * u_hat * dealias
        return self.grid.transform(bih_hat, inverse=True).real

    def hyperviscosity(
        self, u: Array, nu: float, order: int = 2, spectral: bool = False
    ) -> Float[Array, "N"]:
        """Compute the hyperviscosity term: (-1)^{n+1} * nu * d^{2n}u/dx^{2n}.

        This is always dissipative: the spectral multiplier is
        ``-nu * |k|^{2n}``, regardless of *order*.

        Parameters
        ----------
        u : Array [N]
            Input field.
        nu : float
            Hyperviscosity coefficient.
        order : int
            Order n of the hyperviscosity (n=1 is regular diffusion,
            n=2 is biharmonic diffusion). Default: 2.
        spectral : bool
            Whether u is in Fourier space.

        Raises
        ------
        ValueError
            If ``order < 1`` or ``nu < 0``.
        """
        _validate_hyperviscosity(nu, order)
        u_hat = u if spectral else self.grid.transform(u)
        k = self.grid.k
        dealias = self.grid.dealias_filter()
        hyp_hat = -nu * k ** (2 * order) * u_hat * dealias
        return self.grid.transform(hyp_hat, inverse=True).real

    def inverse_laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "N"]:
        """Compute the inverse Laplacian: nabla^{-2} u (1D Poisson solve).

        On a periodic domain, d^2 psi / dx^2 = u has a solution only if u
        has zero spatial mean.  This implementation removes the k=0 component
        of u and returns the unique zero-mean solution psi, i.e. it solves
        d^2 psi / dx^2 = u - mean(u) with psi_hat[k=0] = 0.
        """
        u_hat = u if spectral else self.grid.transform(u)
        k = self.grid.k
        k2 = k**2
        k2_safe = jnp.where(k2 == 0, 1.0, k2)
        psi_hat = -u_hat / k2_safe
        psi_hat = jnp.where(k2 == 0, 0.0, psi_hat)
        return self.grid.transform(psi_hat, inverse=True).real

    def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
        """
        Apply the 2/3 dealiasing rule mask to a field.

        This zeros out the top 1/3 of the spectrum to prevent aliasing
        errors in nonlinear products (e.g., u * du/dx).
        """
        u_hat = u if spectral else self.grid.transform(u)
        mask = self.grid.dealias_filter()
        u_hat_f = u_hat * mask
        return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

Functions

__call__(u, order=1, spectral=False)

Compute the n-th derivative of a field.

Parameters:

u : Array [N] The input field. If spectral=False, this is physical space. If spectral=True, this is complex Fourier coefficients. order : int, optional The order of the derivative (1=first, 2=second, etc.). Default is 1. spectral : bool, optional Whether the input 'u' is already in Fourier space. Default is False.

Returns:

du_dx : Array [N] The n-th derivative in physical space.

Source code in spectraldiffx/_src/fourier/operators.py 
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def __call__(
    self, u: Array, order: int = 1, spectral: bool = False
) -> Float[Array, "N"]:
    """
    Compute the n-th derivative of a field.

    Parameters:
    -----------
    u : Array [N]
        The input field. If spectral=False, this is physical space.
        If spectral=True, this is complex Fourier coefficients.
    order : int, optional
        The order of the derivative (1=first, 2=second, etc.). Default is 1.
    spectral : bool, optional
        Whether the input 'u' is already in Fourier space. Default is False.

    Returns:
    --------
    du_dx : Array [N]
        The n-th derivative in physical space.
    """
    # 1. Obtain spectral coefficients u_hat [N]
    u_hat = u if spectral else self.grid.transform(u)

    # 2. Multiply by (i*k)^order [N]
    # We use k_dealias to zero out high-frequency modes prone to aliasing
    k = self.grid.k_dealias
    du_hat = (1j * k) ** order * u_hat

    # 3. Inverse Transform back to physical space [N]
    return self.grid.transform(du_hat, inverse=True).real

gradient(u, spectral=False)

Compute the first derivative (gradient) du/dx.

Parameters:

u : Array [N] Input field. spectral : bool, optional If True, treats u as Fourier coefficients.

Source code in spectraldiffx/_src/fourier/operators.py 
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def gradient(self, u: Array, spectral: bool = False) -> Array:
    """
    Compute the first derivative (gradient) du/dx.

    Parameters:
    -----------
    u : Array [N]
        Input field.
    spectral : bool, optional
        If True, treats u as Fourier coefficients.
    """
    return self(u, order=1, spectral=spectral)

laplacian(u, spectral=False)

Compute the second derivative (Laplacian) d^2u/dx^2.

Operation in Fourier Space

lap_hat = -(k^2) * u_hat

Source code in spectraldiffx/_src/fourier/operators.py 
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def laplacian(self, u: Array, spectral: bool = False) -> Array:
    """
    Compute the second derivative (Laplacian) d^2u/dx^2.

    Operation in Fourier Space:
        lap_hat = -(k^2) * u_hat
    """
    u_hat = u if spectral else self.grid.transform(u)
    k = self.grid.k
    dealias = self.grid.dealias_filter()
    lap_hat = -(k**2) * u_hat * dealias
    return self.grid.transform(lap_hat, inverse=True).real

biharmonic(u, spectral=False)

Compute the biharmonic operator: d^4u/dx^4.

In Fourier space: k^4 * u_hat.

Source code in spectraldiffx/_src/fourier/operators.py 
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def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "N"]:
    """Compute the biharmonic operator: d^4u/dx^4.

    In Fourier space: k^4 * u_hat.
    """
    u_hat = u if spectral else self.grid.transform(u)
    k = self.grid.k
    dealias = self.grid.dealias_filter()
    bih_hat = k**4 * u_hat * dealias
    return self.grid.transform(bih_hat, inverse=True).real

hyperviscosity(u, nu, order=2, spectral=False)

Compute the hyperviscosity term: (-1)^{n+1} * nu * d^{2n}u/dx^{2n}.

This is always dissipative: the spectral multiplier is -nu * |k|^{2n}, regardless of order.

Parameters

u : Array [N] Input field. nu : float Hyperviscosity coefficient. order : int Order n of the hyperviscosity (n=1 is regular diffusion, n=2 is biharmonic diffusion). Default: 2. spectral : bool Whether u is in Fourier space.

Raises

ValueError If order < 1 or nu < 0.

Source code in spectraldiffx/_src/fourier/operators.py 
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def hyperviscosity(
    self, u: Array, nu: float, order: int = 2, spectral: bool = False
) -> Float[Array, "N"]:
    """Compute the hyperviscosity term: (-1)^{n+1} * nu * d^{2n}u/dx^{2n}.

    This is always dissipative: the spectral multiplier is
    ``-nu * |k|^{2n}``, regardless of *order*.

    Parameters
    ----------
    u : Array [N]
        Input field.
    nu : float
        Hyperviscosity coefficient.
    order : int
        Order n of the hyperviscosity (n=1 is regular diffusion,
        n=2 is biharmonic diffusion). Default: 2.
    spectral : bool
        Whether u is in Fourier space.

    Raises
    ------
    ValueError
        If ``order < 1`` or ``nu < 0``.
    """
    _validate_hyperviscosity(nu, order)
    u_hat = u if spectral else self.grid.transform(u)
    k = self.grid.k
    dealias = self.grid.dealias_filter()
    hyp_hat = -nu * k ** (2 * order) * u_hat * dealias
    return self.grid.transform(hyp_hat, inverse=True).real

inverse_laplacian(u, spectral=False)

Compute the inverse Laplacian: nabla^{-2} u (1D Poisson solve).

On a periodic domain, d^2 psi / dx^2 = u has a solution only if u has zero spatial mean. This implementation removes the k=0 component of u and returns the unique zero-mean solution psi, i.e. it solves d^2 psi / dx^2 = u - mean(u) with psi_hat[k=0] = 0.

Source code in spectraldiffx/_src/fourier/operators.py 
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def inverse_laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "N"]:
    """Compute the inverse Laplacian: nabla^{-2} u (1D Poisson solve).

    On a periodic domain, d^2 psi / dx^2 = u has a solution only if u
    has zero spatial mean.  This implementation removes the k=0 component
    of u and returns the unique zero-mean solution psi, i.e. it solves
    d^2 psi / dx^2 = u - mean(u) with psi_hat[k=0] = 0.
    """
    u_hat = u if spectral else self.grid.transform(u)
    k = self.grid.k
    k2 = k**2
    k2_safe = jnp.where(k2 == 0, 1.0, k2)
    psi_hat = -u_hat / k2_safe
    psi_hat = jnp.where(k2 == 0, 0.0, psi_hat)
    return self.grid.transform(psi_hat, inverse=True).real

apply_dealias(u, spectral=False)

Apply the 2/3 dealiasing rule mask to a field.

This zeros out the top 1/3 of the spectrum to prevent aliasing errors in nonlinear products (e.g., u * du/dx).

Source code in spectraldiffx/_src/fourier/operators.py 
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def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
    """
    Apply the 2/3 dealiasing rule mask to a field.

    This zeros out the top 1/3 of the spectrum to prevent aliasing
    errors in nonlinear products (e.g., u * du/dx).
    """
    u_hat = u if spectral else self.grid.transform(u)
    mask = self.grid.dealias_filter()
    u_hat_f = u_hat * mask
    return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

SpectralDerivative2D

Bases: Module

2D Spectral derivative operators for doubly periodic rectangular domains.

Mathematical Formulation:

For a 2D field u(x, y): u(x, y) = ΣΣ u_hat(kx, ky) * exp(i * (kxx + kyy))

Partial derivatives

∂u/∂x ↔ ikx * u_hat ∂u/∂y ↔ iky * u_hat

Gradient vector: ∇u = (∂u/∂x, ∂u/∂y) Laplacian: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2 ↔ -(kx^2 + ky^2) * u_hat

Source code in spectraldiffx/_src/fourier/operators.py 
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class SpectralDerivative2D(eqx.Module):
    """
    2D Spectral derivative operators for doubly periodic rectangular domains.

    Mathematical Formulation:
    -------------------------
    For a 2D field u(x, y):
        u(x, y) = ΣΣ u_hat(kx, ky) * exp(i * (kx*x + ky*y))

    Partial derivatives:
        ∂u/∂x ↔ i*kx * u_hat
        ∂u/∂y ↔ i*ky * u_hat

    Gradient vector: ∇u = (∂u/∂x, ∂u/∂y)
    Laplacian: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2 ↔ -(kx^2 + ky^2) * u_hat
    """

    grid: FourierGrid2D

    def gradient(
        self, u: Array, spectral: bool = False
    ) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
        """Compute the gradient vector [du/dx, du/dy]."""
        u_hat = u if spectral else self.grid.transform(u)
        KX, KY = self.grid.KX
        dealias = self.grid.dealias_filter()

        du_dx = self.grid.transform(1j * KX * u_hat * dealias, inverse=True).real
        du_dy = self.grid.transform(1j * KY * u_hat * dealias, inverse=True).real
        return du_dx, du_dy

    def divergence(
        self, vx: Array, vy: Array, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """
        Compute the divergence of a 2D vector field: div(V) = ∂vx/∂x + ∂vy/∂y.
        """
        vx_hat = vx if spectral else self.grid.transform(vx)
        vy_hat = vy if spectral else self.grid.transform(vy)
        KX, KY = self.grid.KX
        dealias = self.grid.dealias_filter()

        div_hat = (1j * KX * vx_hat + 1j * KY * vy_hat) * dealias
        return self.grid.transform(div_hat, inverse=True).real

    def curl(
        self, vx: Array, vy: Array, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """
        Compute the 2D scalar curl (vorticity): ζ = ∂vy/∂x - ∂vx/∂y.
        """
        vx_hat = vx if spectral else self.grid.transform(vx)
        vy_hat = vy if spectral else self.grid.transform(vy)
        KX, KY = self.grid.KX
        dealias = self.grid.dealias_filter()

        curl_hat = (1j * KX * vy_hat - 1j * KY * vx_hat) * dealias
        return self.grid.transform(curl_hat, inverse=True).real

    def laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "Ny Nx"]:
        """
        Compute the 2D Laplacian: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2.
        """
        u_hat = u if spectral else self.grid.transform(u)
        K2 = self.grid.K2
        dealias = self.grid.dealias_filter()
        lap_hat = -K2 * u_hat * dealias
        return self.grid.transform(lap_hat, inverse=True).real

    def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "Ny Nx"]:
        """Compute the 2D biharmonic: nabla^4 u = (kx^2 + ky^2)^2 * u_hat."""
        u_hat = u if spectral else self.grid.transform(u)
        K2 = self.grid.K2
        dealias = self.grid.dealias_filter()
        bih_hat = K2**2 * u_hat * dealias
        return self.grid.transform(bih_hat, inverse=True).real

    def hyperviscosity(
        self, u: Array, nu: float, order: int = 2, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """Compute the 2D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

        This is always dissipative: the spectral multiplier is
        ``-nu * |k|^{2n}``, regardless of *order*.

        Parameters
        ----------
        u : Array [Ny, Nx]
            Input field.
        nu : float
            Hyperviscosity coefficient.
        order : int
            Order n (n=1 is diffusion, n=2 is biharmonic). Default: 2.
        spectral : bool
            Whether u is in Fourier space.

        Raises
        ------
        ValueError
            If ``order < 1`` or ``nu < 0``.
        """
        _validate_hyperviscosity(nu, order)
        u_hat = u if spectral else self.grid.transform(u)
        K2 = self.grid.K2
        dealias = self.grid.dealias_filter()
        hyp_hat = -nu * K2**order * u_hat * dealias
        return self.grid.transform(hyp_hat, inverse=True).real

    def inverse_laplacian(
        self, u: Array, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """Compute the 2D inverse Laplacian: nabla^{-2} u (Poisson solve).

        On a periodic domain, nabla^2 psi = u is solvable only if u has zero
        spatial mean.  This implementation removes the (0,0) Fourier mode
        of u and returns the unique zero-mean solution psi.
        """
        u_hat = u if spectral else self.grid.transform(u)
        K2 = self.grid.K2
        K2_safe = jnp.where(K2 == 0, 1.0, K2)
        psi_hat = -u_hat / K2_safe
        psi_hat = jnp.where(K2 == 0, 0.0, psi_hat)
        return self.grid.transform(psi_hat, inverse=True).real

    def velocity_from_streamfunction(
        self, psi: Array, spectral: bool = False
    ) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
        """Compute velocity from a streamfunction: u = -dpsi/dy, v = dpsi/dx."""
        psi_hat = psi if spectral else self.grid.transform(psi)
        KX, KY = self.grid.KX
        dealias = self.grid.dealias_filter()
        u = self.grid.transform(-1j * KY * psi_hat * dealias, inverse=True).real
        v = self.grid.transform(1j * KX * psi_hat * dealias, inverse=True).real
        return u, v

    def jacobian(
        self, f: Array, g: Array, spectral: bool = False
    ) -> Float[Array, "Ny Nx"]:
        """Compute the Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

        Uses the pseudo-spectral method: derivatives in Fourier space,
        products in physical space, with dealiasing.
        """
        f_hat = f if spectral else self.grid.transform(f)
        g_hat = g if spectral else self.grid.transform(g)
        KX, KY = self.grid.KX
        dealias = self.grid.dealias_filter()

        df_dx = self.grid.transform(1j * KX * f_hat * dealias, inverse=True).real
        df_dy = self.grid.transform(1j * KY * f_hat * dealias, inverse=True).real
        dg_dx = self.grid.transform(1j * KX * g_hat * dealias, inverse=True).real
        dg_dy = self.grid.transform(1j * KY * g_hat * dealias, inverse=True).real

        jac = df_dx * dg_dy - df_dy * dg_dx
        return self.apply_dealias(jac)

    def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
        """Apply the 2D spectral dealiasing filter mask [Ny, Nx]."""
        u_hat = u if spectral else self.grid.transform(u)
        mask = self.grid.dealias_filter()
        u_hat_f = u_hat * mask
        return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

    def project_vector(self, vx: Array, vy: Array) -> tuple[Array, Array]:
        """
        Perform Leray projection to extract the divergence-free component.

        Solves the decomposition: V = V_solenoidal + V_irrotational
        where div(V_solenoidal) = 0 and curl(V_irrotational) = 0.

        Physics:
        --------
        In incompressible flows, we project the velocity onto the divergence-free
        manifold by solving for a scalar potential φ:
            div(V - grad(φ)) = 0  =>  ∇^2 φ = div(V)

        Then: V_solenoidal = V - grad(φ)
        """
        # 1. Transform components to Fourier space [Ny, Nx]
        vx_hat = self.grid.transform(vx)
        vy_hat = self.grid.transform(vy)
        KX, KY = self.grid.KX

        # 2. Compute divergence in Fourier space
        div_hat = 1j * (KX * vx_hat + KY * vy_hat)

        # 3. Solve Poisson: -|k|^2 * φ_hat = div_hat
        K2 = self.grid.K2
        K2_safe = jnp.where(K2 == 0, 1.0, K2)
        phi_hat = -div_hat / K2_safe
        phi_hat = jnp.where(K2 == 0, 0.0, phi_hat)  # Force mean to zero

        # 4. Correct velocity: V_sol_hat = V_hat - i*k*φ_hat
        vx_sol_hat = vx_hat - 1j * KX * phi_hat
        vy_sol_hat = vy_hat - 1j * KY * phi_hat

        return (
            self.grid.transform(vx_sol_hat, inverse=True).real,
            self.grid.transform(vy_sol_hat, inverse=True).real,
        )

    def advection_scalar(self, vx: Array, vy: Array, q: Array) -> Float[Array, "Ny Nx"]:
        """
        Compute scalar advection (u·∇)q using the pseudo-spectral method.

        This method computes derivatives in Fourier space for accuracy, then
        transforms back to physical space to perform the multiplication.

        Out = vx * ∂q/∂x + vy * ∂q/∂y
        """
        q_hat = self.grid.transform(q)
        KX, KY = self.grid.KX
        dealias = self.grid.dealias_filter()

        dq_dx = self.grid.transform(1j * KX * q_hat * dealias, inverse=True).real
        dq_dy = self.grid.transform(1j * KY * q_hat * dealias, inverse=True).real

        return vx * dq_dx + vy * dq_dy

Functions

gradient(u, spectral=False)

Compute the gradient vector [du/dx, du/dy].

Source code in spectraldiffx/_src/fourier/operators.py 
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def gradient(
    self, u: Array, spectral: bool = False
) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
    """Compute the gradient vector [du/dx, du/dy]."""
    u_hat = u if spectral else self.grid.transform(u)
    KX, KY = self.grid.KX
    dealias = self.grid.dealias_filter()

    du_dx = self.grid.transform(1j * KX * u_hat * dealias, inverse=True).real
    du_dy = self.grid.transform(1j * KY * u_hat * dealias, inverse=True).real
    return du_dx, du_dy

divergence(vx, vy, spectral=False)

Compute the divergence of a 2D vector field: div(V) = ∂vx/∂x + ∂vy/∂y.

Source code in spectraldiffx/_src/fourier/operators.py 
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def divergence(
    self, vx: Array, vy: Array, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """
    Compute the divergence of a 2D vector field: div(V) = ∂vx/∂x + ∂vy/∂y.
    """
    vx_hat = vx if spectral else self.grid.transform(vx)
    vy_hat = vy if spectral else self.grid.transform(vy)
    KX, KY = self.grid.KX
    dealias = self.grid.dealias_filter()

    div_hat = (1j * KX * vx_hat + 1j * KY * vy_hat) * dealias
    return self.grid.transform(div_hat, inverse=True).real

curl(vx, vy, spectral=False)

Compute the 2D scalar curl (vorticity): ζ = ∂vy/∂x - ∂vx/∂y.

Source code in spectraldiffx/_src/fourier/operators.py 
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def curl(
    self, vx: Array, vy: Array, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """
    Compute the 2D scalar curl (vorticity): ζ = ∂vy/∂x - ∂vx/∂y.
    """
    vx_hat = vx if spectral else self.grid.transform(vx)
    vy_hat = vy if spectral else self.grid.transform(vy)
    KX, KY = self.grid.KX
    dealias = self.grid.dealias_filter()

    curl_hat = (1j * KX * vy_hat - 1j * KY * vx_hat) * dealias
    return self.grid.transform(curl_hat, inverse=True).real

laplacian(u, spectral=False)

Compute the 2D Laplacian: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2.

Source code in spectraldiffx/_src/fourier/operators.py 
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def laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "Ny Nx"]:
    """
    Compute the 2D Laplacian: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2.
    """
    u_hat = u if spectral else self.grid.transform(u)
    K2 = self.grid.K2
    dealias = self.grid.dealias_filter()
    lap_hat = -K2 * u_hat * dealias
    return self.grid.transform(lap_hat, inverse=True).real

biharmonic(u, spectral=False)

Compute the 2D biharmonic: nabla^4 u = (kx^2 + ky^2)^2 * u_hat.

Source code in spectraldiffx/_src/fourier/operators.py 
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def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "Ny Nx"]:
    """Compute the 2D biharmonic: nabla^4 u = (kx^2 + ky^2)^2 * u_hat."""
    u_hat = u if spectral else self.grid.transform(u)
    K2 = self.grid.K2
    dealias = self.grid.dealias_filter()
    bih_hat = K2**2 * u_hat * dealias
    return self.grid.transform(bih_hat, inverse=True).real

hyperviscosity(u, nu, order=2, spectral=False)

Compute the 2D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

This is always dissipative: the spectral multiplier is -nu * |k|^{2n}, regardless of order.

Parameters

u : Array [Ny, Nx] Input field. nu : float Hyperviscosity coefficient. order : int Order n (n=1 is diffusion, n=2 is biharmonic). Default: 2. spectral : bool Whether u is in Fourier space.

Raises

ValueError If order < 1 or nu < 0.

Source code in spectraldiffx/_src/fourier/operators.py 
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def hyperviscosity(
    self, u: Array, nu: float, order: int = 2, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """Compute the 2D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

    This is always dissipative: the spectral multiplier is
    ``-nu * |k|^{2n}``, regardless of *order*.

    Parameters
    ----------
    u : Array [Ny, Nx]
        Input field.
    nu : float
        Hyperviscosity coefficient.
    order : int
        Order n (n=1 is diffusion, n=2 is biharmonic). Default: 2.
    spectral : bool
        Whether u is in Fourier space.

    Raises
    ------
    ValueError
        If ``order < 1`` or ``nu < 0``.
    """
    _validate_hyperviscosity(nu, order)
    u_hat = u if spectral else self.grid.transform(u)
    K2 = self.grid.K2
    dealias = self.grid.dealias_filter()
    hyp_hat = -nu * K2**order * u_hat * dealias
    return self.grid.transform(hyp_hat, inverse=True).real

inverse_laplacian(u, spectral=False)

Compute the 2D inverse Laplacian: nabla^{-2} u (Poisson solve).

On a periodic domain, nabla^2 psi = u is solvable only if u has zero spatial mean. This implementation removes the (0,0) Fourier mode of u and returns the unique zero-mean solution psi.

Source code in spectraldiffx/_src/fourier/operators.py 
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def inverse_laplacian(
    self, u: Array, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """Compute the 2D inverse Laplacian: nabla^{-2} u (Poisson solve).

    On a periodic domain, nabla^2 psi = u is solvable only if u has zero
    spatial mean.  This implementation removes the (0,0) Fourier mode
    of u and returns the unique zero-mean solution psi.
    """
    u_hat = u if spectral else self.grid.transform(u)
    K2 = self.grid.K2
    K2_safe = jnp.where(K2 == 0, 1.0, K2)
    psi_hat = -u_hat / K2_safe
    psi_hat = jnp.where(K2 == 0, 0.0, psi_hat)
    return self.grid.transform(psi_hat, inverse=True).real

velocity_from_streamfunction(psi, spectral=False)

Compute velocity from a streamfunction: u = -dpsi/dy, v = dpsi/dx.

Source code in spectraldiffx/_src/fourier/operators.py 
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def velocity_from_streamfunction(
    self, psi: Array, spectral: bool = False
) -> tuple[Float[Array, "Ny Nx"], Float[Array, "Ny Nx"]]:
    """Compute velocity from a streamfunction: u = -dpsi/dy, v = dpsi/dx."""
    psi_hat = psi if spectral else self.grid.transform(psi)
    KX, KY = self.grid.KX
    dealias = self.grid.dealias_filter()
    u = self.grid.transform(-1j * KY * psi_hat * dealias, inverse=True).real
    v = self.grid.transform(1j * KX * psi_hat * dealias, inverse=True).real
    return u, v

jacobian(f, g, spectral=False)

Compute the Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

Uses the pseudo-spectral method: derivatives in Fourier space, products in physical space, with dealiasing.

Source code in spectraldiffx/_src/fourier/operators.py 
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def jacobian(
    self, f: Array, g: Array, spectral: bool = False
) -> Float[Array, "Ny Nx"]:
    """Compute the Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

    Uses the pseudo-spectral method: derivatives in Fourier space,
    products in physical space, with dealiasing.
    """
    f_hat = f if spectral else self.grid.transform(f)
    g_hat = g if spectral else self.grid.transform(g)
    KX, KY = self.grid.KX
    dealias = self.grid.dealias_filter()

    df_dx = self.grid.transform(1j * KX * f_hat * dealias, inverse=True).real
    df_dy = self.grid.transform(1j * KY * f_hat * dealias, inverse=True).real
    dg_dx = self.grid.transform(1j * KX * g_hat * dealias, inverse=True).real
    dg_dy = self.grid.transform(1j * KY * g_hat * dealias, inverse=True).real

    jac = df_dx * dg_dy - df_dy * dg_dx
    return self.apply_dealias(jac)

apply_dealias(u, spectral=False)

Apply the 2D spectral dealiasing filter mask [Ny, Nx].

Source code in spectraldiffx/_src/fourier/operators.py 
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def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
    """Apply the 2D spectral dealiasing filter mask [Ny, Nx]."""
    u_hat = u if spectral else self.grid.transform(u)
    mask = self.grid.dealias_filter()
    u_hat_f = u_hat * mask
    return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

project_vector(vx, vy)

Perform Leray projection to extract the divergence-free component.

Solves the decomposition: V = V_solenoidal + V_irrotational where div(V_solenoidal) = 0 and curl(V_irrotational) = 0.

Physics:

In incompressible flows, we project the velocity onto the divergence-free manifold by solving for a scalar potential φ: div(V - grad(φ)) = 0 => ∇^2 φ = div(V)

Then: V_solenoidal = V - grad(φ)

Source code in spectraldiffx/_src/fourier/operators.py 
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def project_vector(self, vx: Array, vy: Array) -> tuple[Array, Array]:
    """
    Perform Leray projection to extract the divergence-free component.

    Solves the decomposition: V = V_solenoidal + V_irrotational
    where div(V_solenoidal) = 0 and curl(V_irrotational) = 0.

    Physics:
    --------
    In incompressible flows, we project the velocity onto the divergence-free
    manifold by solving for a scalar potential φ:
        div(V - grad(φ)) = 0  =>  ∇^2 φ = div(V)

    Then: V_solenoidal = V - grad(φ)
    """
    # 1. Transform components to Fourier space [Ny, Nx]
    vx_hat = self.grid.transform(vx)
    vy_hat = self.grid.transform(vy)
    KX, KY = self.grid.KX

    # 2. Compute divergence in Fourier space
    div_hat = 1j * (KX * vx_hat + KY * vy_hat)

    # 3. Solve Poisson: -|k|^2 * φ_hat = div_hat
    K2 = self.grid.K2
    K2_safe = jnp.where(K2 == 0, 1.0, K2)
    phi_hat = -div_hat / K2_safe
    phi_hat = jnp.where(K2 == 0, 0.0, phi_hat)  # Force mean to zero

    # 4. Correct velocity: V_sol_hat = V_hat - i*k*φ_hat
    vx_sol_hat = vx_hat - 1j * KX * phi_hat
    vy_sol_hat = vy_hat - 1j * KY * phi_hat

    return (
        self.grid.transform(vx_sol_hat, inverse=True).real,
        self.grid.transform(vy_sol_hat, inverse=True).real,
    )

advection_scalar(vx, vy, q)

Compute scalar advection (u·∇)q using the pseudo-spectral method.

This method computes derivatives in Fourier space for accuracy, then transforms back to physical space to perform the multiplication.

Out = vx * ∂q/∂x + vy * ∂q/∂y

Source code in spectraldiffx/_src/fourier/operators.py 
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def advection_scalar(self, vx: Array, vy: Array, q: Array) -> Float[Array, "Ny Nx"]:
    """
    Compute scalar advection (u·∇)q using the pseudo-spectral method.

    This method computes derivatives in Fourier space for accuracy, then
    transforms back to physical space to perform the multiplication.

    Out = vx * ∂q/∂x + vy * ∂q/∂y
    """
    q_hat = self.grid.transform(q)
    KX, KY = self.grid.KX
    dealias = self.grid.dealias_filter()

    dq_dx = self.grid.transform(1j * KX * q_hat * dealias, inverse=True).real
    dq_dy = self.grid.transform(1j * KY * q_hat * dealias, inverse=True).real

    return vx * dq_dx + vy * dq_dy

SpectralDerivative3D

Bases: Module

3D Spectral derivative operators for triply periodic domains.

Mathematical Framework:

Field shapes: (Nz, Ny, Nx) following 'ij' indexing. Wavenumbers: (kz, ky, kx).

Gradient vector: ∇u = (∂u/∂z, ∂u/∂y, ∂u/∂x)

Source code in spectraldiffx/_src/fourier/operators.py 
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class SpectralDerivative3D(eqx.Module):
    """
    3D Spectral derivative operators for triply periodic domains.

    Mathematical Framework:
    -----------------------
    Field shapes: (Nz, Ny, Nx) following 'ij' indexing.
    Wavenumbers: (kz, ky, kx).

    Gradient vector: ∇u = (∂u/∂z, ∂u/∂y, ∂u/∂x)
    """

    grid: FourierGrid3D

    def gradient(self, u: Array, spectral: bool = False) -> tuple[Array, Array, Array]:
        """Compute the 3D gradient vector field."""
        u_hat = u if spectral else self.grid.transform(u)
        KZ, KY, KX = self.grid.KX
        dealias = self.grid.dealias_filter()

        du_dz = self.grid.transform(1j * KZ * u_hat * dealias, inverse=True).real
        du_dy = self.grid.transform(1j * KY * u_hat * dealias, inverse=True).real
        du_dx = self.grid.transform(1j * KX * u_hat * dealias, inverse=True).real
        return du_dz, du_dy, du_dx

    def divergence(
        self, vz: Array, vy: Array, vx: Array, spectral: bool = False
    ) -> Float[Array, "Nz Ny Nx"]:
        """Compute 3D divergence: ∂vz/∂z + ∂vy/∂y + ∂vx/∂x."""
        vz_hat = vz if spectral else self.grid.transform(vz)
        vy_hat = vy if spectral else self.grid.transform(vy)
        vx_hat = vx if spectral else self.grid.transform(vx)
        KZ, KY, KX = self.grid.KX
        dealias = self.grid.dealias_filter()

        div_hat = (1j * KZ * vz_hat + 1j * KY * vy_hat + 1j * KX * vx_hat) * dealias
        return self.grid.transform(div_hat, inverse=True).real

    def curl(
        self, vz: Array, vy: Array, vx: Array, spectral: bool = False
    ) -> tuple[Array, Array, Array]:
        """
        Compute the 3D curl vector: ω = ∇ x V.

        Components:
            ωz = ∂vy/∂x - ∂vx/∂y
            ωy = ∂vx/∂z - ∂vz/∂x
            ωx = ∂vz/∂y - ∂vy/∂z
        """
        vz_hat = vz if spectral else self.grid.transform(vz)
        vy_hat = vy if spectral else self.grid.transform(vy)
        vx_hat = vx if spectral else self.grid.transform(vx)
        KZ, KY, KX = self.grid.KX
        dealias = self.grid.dealias_filter()

        wz_hat = (1j * KX * vy_hat - 1j * KY * vx_hat) * dealias
        wy_hat = (1j * KZ * vx_hat - 1j * KX * vz_hat) * dealias
        wx_hat = (1j * KY * vz_hat - 1j * KZ * vy_hat) * dealias

        return (
            self.grid.transform(wz_hat, inverse=True).real,
            self.grid.transform(wy_hat, inverse=True).real,
            self.grid.transform(wx_hat, inverse=True).real,
        )

    def laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "Nz Ny Nx"]:
        """Compute 3D Laplacian: ∇^2 u = ∂^2u/∂z^2 + ∂^2u/∂y^2 + ∂^2u/∂x^2."""
        u_hat = u if spectral else self.grid.transform(u)
        K2 = self.grid.K2
        dealias = self.grid.dealias_filter()
        lap_hat = -K2 * u_hat * dealias
        return self.grid.transform(lap_hat, inverse=True).real

    def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "Nz Ny Nx"]:
        """Compute the 3D biharmonic: nabla^4 u."""
        u_hat = u if spectral else self.grid.transform(u)
        K2 = self.grid.K2
        dealias = self.grid.dealias_filter()
        bih_hat = K2**2 * u_hat * dealias
        return self.grid.transform(bih_hat, inverse=True).real

    def hyperviscosity(
        self, u: Array, nu: float, order: int = 2, spectral: bool = False
    ) -> Float[Array, "Nz Ny Nx"]:
        """Compute the 3D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

        Raises
        ------
        ValueError
            If ``order < 1`` or ``nu < 0``.
        """
        _validate_hyperviscosity(nu, order)
        u_hat = u if spectral else self.grid.transform(u)
        K2 = self.grid.K2
        dealias = self.grid.dealias_filter()
        hyp_hat = -nu * K2**order * u_hat * dealias
        return self.grid.transform(hyp_hat, inverse=True).real

    def inverse_laplacian(
        self, u: Array, spectral: bool = False
    ) -> Float[Array, "Nz Ny Nx"]:
        """Compute the 3D inverse Laplacian: nabla^{-2} u (Poisson solve).

        On a periodic domain, nabla^2 psi = u is solvable only if u has zero
        spatial mean.  This implementation removes the (0,0,0) Fourier mode
        of u and returns the unique zero-mean solution psi.
        """
        u_hat = u if spectral else self.grid.transform(u)
        K2 = self.grid.K2
        K2_safe = jnp.where(K2 == 0, 1.0, K2)
        psi_hat = -u_hat / K2_safe
        psi_hat = jnp.where(K2 == 0, 0.0, psi_hat)
        return self.grid.transform(psi_hat, inverse=True).real

    def velocity_from_streamfunction(
        self, psi: Array, spectral: bool = False
    ) -> tuple[Float[Array, "Nz Ny Nx"], Float[Array, "Nz Ny Nx"]]:
        """Compute horizontal velocity from a streamfunction.

        Returns (u, v) where u = -dpsi/dy, v = dpsi/dx (horizontal components).
        """
        psi_hat = psi if spectral else self.grid.transform(psi)
        _KZ, KY, KX = self.grid.KX
        dealias = self.grid.dealias_filter()
        u = self.grid.transform(-1j * KY * psi_hat * dealias, inverse=True).real
        v = self.grid.transform(1j * KX * psi_hat * dealias, inverse=True).real
        return u, v

    def jacobian(
        self, f: Array, g: Array, spectral: bool = False
    ) -> Float[Array, "Nz Ny Nx"]:
        """Compute the horizontal Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

        Operates on the horizontal (x, y) components only — the z direction
        is treated as a batch dimension.
        """
        f_hat = f if spectral else self.grid.transform(f)
        g_hat = g if spectral else self.grid.transform(g)
        _KZ, KY, KX = self.grid.KX
        dealias = self.grid.dealias_filter()

        df_dx = self.grid.transform(1j * KX * f_hat * dealias, inverse=True).real
        df_dy = self.grid.transform(1j * KY * f_hat * dealias, inverse=True).real
        dg_dx = self.grid.transform(1j * KX * g_hat * dealias, inverse=True).real
        dg_dy = self.grid.transform(1j * KY * g_hat * dealias, inverse=True).real

        jac = df_dx * dg_dy - df_dy * dg_dx
        return self.apply_dealias(jac)

    def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
        """Apply the 3D periodic dealiasing filter mask [Nz, Ny, Nx]."""
        u_hat = u if spectral else self.grid.transform(u)
        mask = self.grid.dealias_filter()
        u_hat_f = u_hat * mask
        return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

    def project_vector(
        self, vz: Array, vy: Array, vx: Array
    ) -> tuple[Array, Array, Array]:
        """Project a 3D vector field onto its solenoidal (divergence-free) component."""
        vz_hat = self.grid.transform(vz)
        vy_hat = self.grid.transform(vy)
        vx_hat = self.grid.transform(vx)
        KZ, KY, KX = self.grid.KX

        div_hat = 1j * (KZ * vz_hat + KY * vy_hat + KX * vx_hat)

        K2 = self.grid.K2
        K2_safe = jnp.where(K2 == 0, 1.0, K2)
        phi_hat = -div_hat / K2_safe
        phi_hat = jnp.where(K2 == 0, 0.0, phi_hat)

        vz_sol_hat = vz_hat - 1j * KZ * phi_hat
        vy_sol_hat = vy_hat - 1j * KY * phi_hat
        vx_sol_hat = vx_hat - 1j * KX * phi_hat

        return (
            self.grid.transform(vz_sol_hat, inverse=True).real,
            self.grid.transform(vy_sol_hat, inverse=True).real,
            self.grid.transform(vx_sol_hat, inverse=True).real,
        )

    def advection_scalar(
        self, vz: Array, vy: Array, vx: Array, q: Array
    ) -> Float[Array, "Nz Ny Nx"]:
        """Compute the 3D scalar advection term: (u·∇)q."""
        q_hat = self.grid.transform(q)
        KZ, KY, KX = self.grid.KX
        dealias = self.grid.dealias_filter()

        dq_dz = self.grid.transform(1j * KZ * q_hat * dealias, inverse=True).real
        dq_dy = self.grid.transform(1j * KY * q_hat * dealias, inverse=True).real
        dq_dx = self.grid.transform(1j * KX * q_hat * dealias, inverse=True).real

        return vz * dq_dz + vy * dq_dy + vx * dq_dx

Functions

gradient(u, spectral=False)

Compute the 3D gradient vector field.

Source code in spectraldiffx/_src/fourier/operators.py 
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def gradient(self, u: Array, spectral: bool = False) -> tuple[Array, Array, Array]:
    """Compute the 3D gradient vector field."""
    u_hat = u if spectral else self.grid.transform(u)
    KZ, KY, KX = self.grid.KX
    dealias = self.grid.dealias_filter()

    du_dz = self.grid.transform(1j * KZ * u_hat * dealias, inverse=True).real
    du_dy = self.grid.transform(1j * KY * u_hat * dealias, inverse=True).real
    du_dx = self.grid.transform(1j * KX * u_hat * dealias, inverse=True).real
    return du_dz, du_dy, du_dx

divergence(vz, vy, vx, spectral=False)

Compute 3D divergence: ∂vz/∂z + ∂vy/∂y + ∂vx/∂x.

Source code in spectraldiffx/_src/fourier/operators.py 
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def divergence(
    self, vz: Array, vy: Array, vx: Array, spectral: bool = False
) -> Float[Array, "Nz Ny Nx"]:
    """Compute 3D divergence: ∂vz/∂z + ∂vy/∂y + ∂vx/∂x."""
    vz_hat = vz if spectral else self.grid.transform(vz)
    vy_hat = vy if spectral else self.grid.transform(vy)
    vx_hat = vx if spectral else self.grid.transform(vx)
    KZ, KY, KX = self.grid.KX
    dealias = self.grid.dealias_filter()

    div_hat = (1j * KZ * vz_hat + 1j * KY * vy_hat + 1j * KX * vx_hat) * dealias
    return self.grid.transform(div_hat, inverse=True).real

curl(vz, vy, vx, spectral=False)

Compute the 3D curl vector: ω = ∇ x V.

Components

ωz = ∂vy/∂x - ∂vx/∂y ωy = ∂vx/∂z - ∂vz/∂x ωx = ∂vz/∂y - ∂vy/∂z

Source code in spectraldiffx/_src/fourier/operators.py 
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def curl(
    self, vz: Array, vy: Array, vx: Array, spectral: bool = False
) -> tuple[Array, Array, Array]:
    """
    Compute the 3D curl vector: ω = ∇ x V.

    Components:
        ωz = ∂vy/∂x - ∂vx/∂y
        ωy = ∂vx/∂z - ∂vz/∂x
        ωx = ∂vz/∂y - ∂vy/∂z
    """
    vz_hat = vz if spectral else self.grid.transform(vz)
    vy_hat = vy if spectral else self.grid.transform(vy)
    vx_hat = vx if spectral else self.grid.transform(vx)
    KZ, KY, KX = self.grid.KX
    dealias = self.grid.dealias_filter()

    wz_hat = (1j * KX * vy_hat - 1j * KY * vx_hat) * dealias
    wy_hat = (1j * KZ * vx_hat - 1j * KX * vz_hat) * dealias
    wx_hat = (1j * KY * vz_hat - 1j * KZ * vy_hat) * dealias

    return (
        self.grid.transform(wz_hat, inverse=True).real,
        self.grid.transform(wy_hat, inverse=True).real,
        self.grid.transform(wx_hat, inverse=True).real,
    )

laplacian(u, spectral=False)

Compute 3D Laplacian: ∇^2 u = ∂^2u/∂z^2 + ∂^2u/∂y^2 + ∂^2u/∂x^2.

Source code in spectraldiffx/_src/fourier/operators.py 
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def laplacian(self, u: Array, spectral: bool = False) -> Float[Array, "Nz Ny Nx"]:
    """Compute 3D Laplacian: ∇^2 u = ∂^2u/∂z^2 + ∂^2u/∂y^2 + ∂^2u/∂x^2."""
    u_hat = u if spectral else self.grid.transform(u)
    K2 = self.grid.K2
    dealias = self.grid.dealias_filter()
    lap_hat = -K2 * u_hat * dealias
    return self.grid.transform(lap_hat, inverse=True).real

biharmonic(u, spectral=False)

Compute the 3D biharmonic: nabla^4 u.

Source code in spectraldiffx/_src/fourier/operators.py 
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def biharmonic(self, u: Array, spectral: bool = False) -> Float[Array, "Nz Ny Nx"]:
    """Compute the 3D biharmonic: nabla^4 u."""
    u_hat = u if spectral else self.grid.transform(u)
    K2 = self.grid.K2
    dealias = self.grid.dealias_filter()
    bih_hat = K2**2 * u_hat * dealias
    return self.grid.transform(bih_hat, inverse=True).real

hyperviscosity(u, nu, order=2, spectral=False)

Compute the 3D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

Raises

ValueError If order < 1 or nu < 0.

Source code in spectraldiffx/_src/fourier/operators.py 
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def hyperviscosity(
    self, u: Array, nu: float, order: int = 2, spectral: bool = False
) -> Float[Array, "Nz Ny Nx"]:
    """Compute the 3D hyperviscosity: (-1)^{n+1} * nu * nabla^{2n} u.

    Raises
    ------
    ValueError
        If ``order < 1`` or ``nu < 0``.
    """
    _validate_hyperviscosity(nu, order)
    u_hat = u if spectral else self.grid.transform(u)
    K2 = self.grid.K2
    dealias = self.grid.dealias_filter()
    hyp_hat = -nu * K2**order * u_hat * dealias
    return self.grid.transform(hyp_hat, inverse=True).real

inverse_laplacian(u, spectral=False)

Compute the 3D inverse Laplacian: nabla^{-2} u (Poisson solve).

On a periodic domain, nabla^2 psi = u is solvable only if u has zero spatial mean. This implementation removes the (0,0,0) Fourier mode of u and returns the unique zero-mean solution psi.

Source code in spectraldiffx/_src/fourier/operators.py 
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def inverse_laplacian(
    self, u: Array, spectral: bool = False
) -> Float[Array, "Nz Ny Nx"]:
    """Compute the 3D inverse Laplacian: nabla^{-2} u (Poisson solve).

    On a periodic domain, nabla^2 psi = u is solvable only if u has zero
    spatial mean.  This implementation removes the (0,0,0) Fourier mode
    of u and returns the unique zero-mean solution psi.
    """
    u_hat = u if spectral else self.grid.transform(u)
    K2 = self.grid.K2
    K2_safe = jnp.where(K2 == 0, 1.0, K2)
    psi_hat = -u_hat / K2_safe
    psi_hat = jnp.where(K2 == 0, 0.0, psi_hat)
    return self.grid.transform(psi_hat, inverse=True).real

velocity_from_streamfunction(psi, spectral=False)

Compute horizontal velocity from a streamfunction.

Returns (u, v) where u = -dpsi/dy, v = dpsi/dx (horizontal components).

Source code in spectraldiffx/_src/fourier/operators.py 
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def velocity_from_streamfunction(
    self, psi: Array, spectral: bool = False
) -> tuple[Float[Array, "Nz Ny Nx"], Float[Array, "Nz Ny Nx"]]:
    """Compute horizontal velocity from a streamfunction.

    Returns (u, v) where u = -dpsi/dy, v = dpsi/dx (horizontal components).
    """
    psi_hat = psi if spectral else self.grid.transform(psi)
    _KZ, KY, KX = self.grid.KX
    dealias = self.grid.dealias_filter()
    u = self.grid.transform(-1j * KY * psi_hat * dealias, inverse=True).real
    v = self.grid.transform(1j * KX * psi_hat * dealias, inverse=True).real
    return u, v

jacobian(f, g, spectral=False)

Compute the horizontal Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

Operates on the horizontal (x, y) components only — the z direction is treated as a batch dimension.

Source code in spectraldiffx/_src/fourier/operators.py 
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def jacobian(
    self, f: Array, g: Array, spectral: bool = False
) -> Float[Array, "Nz Ny Nx"]:
    """Compute the horizontal Jacobian: J(f,g) = df/dx * dg/dy - df/dy * dg/dx.

    Operates on the horizontal (x, y) components only — the z direction
    is treated as a batch dimension.
    """
    f_hat = f if spectral else self.grid.transform(f)
    g_hat = g if spectral else self.grid.transform(g)
    _KZ, KY, KX = self.grid.KX
    dealias = self.grid.dealias_filter()

    df_dx = self.grid.transform(1j * KX * f_hat * dealias, inverse=True).real
    df_dy = self.grid.transform(1j * KY * f_hat * dealias, inverse=True).real
    dg_dx = self.grid.transform(1j * KX * g_hat * dealias, inverse=True).real
    dg_dy = self.grid.transform(1j * KY * g_hat * dealias, inverse=True).real

    jac = df_dx * dg_dy - df_dy * dg_dx
    return self.apply_dealias(jac)

apply_dealias(u, spectral=False)

Apply the 3D periodic dealiasing filter mask [Nz, Ny, Nx].

Source code in spectraldiffx/_src/fourier/operators.py 
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def apply_dealias(self, u: Array, spectral: bool = False) -> Array:
    """Apply the 3D periodic dealiasing filter mask [Nz, Ny, Nx]."""
    u_hat = u if spectral else self.grid.transform(u)
    mask = self.grid.dealias_filter()
    u_hat_f = u_hat * mask
    return u_hat_f if spectral else self.grid.transform(u_hat_f, inverse=True).real

project_vector(vz, vy, vx)

Project a 3D vector field onto its solenoidal (divergence-free) component.

Source code in spectraldiffx/_src/fourier/operators.py 
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def project_vector(
    self, vz: Array, vy: Array, vx: Array
) -> tuple[Array, Array, Array]:
    """Project a 3D vector field onto its solenoidal (divergence-free) component."""
    vz_hat = self.grid.transform(vz)
    vy_hat = self.grid.transform(vy)
    vx_hat = self.grid.transform(vx)
    KZ, KY, KX = self.grid.KX

    div_hat = 1j * (KZ * vz_hat + KY * vy_hat + KX * vx_hat)

    K2 = self.grid.K2
    K2_safe = jnp.where(K2 == 0, 1.0, K2)
    phi_hat = -div_hat / K2_safe
    phi_hat = jnp.where(K2 == 0, 0.0, phi_hat)

    vz_sol_hat = vz_hat - 1j * KZ * phi_hat
    vy_sol_hat = vy_hat - 1j * KY * phi_hat
    vx_sol_hat = vx_hat - 1j * KX * phi_hat

    return (
        self.grid.transform(vz_sol_hat, inverse=True).real,
        self.grid.transform(vy_sol_hat, inverse=True).real,
        self.grid.transform(vx_sol_hat, inverse=True).real,
    )

advection_scalar(vz, vy, vx, q)

Compute the 3D scalar advection term: (u·∇)q.

Source code in spectraldiffx/_src/fourier/operators.py 
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def advection_scalar(
    self, vz: Array, vy: Array, vx: Array, q: Array
) -> Float[Array, "Nz Ny Nx"]:
    """Compute the 3D scalar advection term: (u·∇)q."""
    q_hat = self.grid.transform(q)
    KZ, KY, KX = self.grid.KX
    dealias = self.grid.dealias_filter()

    dq_dz = self.grid.transform(1j * KZ * q_hat * dealias, inverse=True).real
    dq_dy = self.grid.transform(1j * KY * q_hat * dealias, inverse=True).real
    dq_dx = self.grid.transform(1j * KX * q_hat * dealias, inverse=True).real

    return vz * dq_dz + vy * dq_dy + vx * dq_dx