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Hyper-Diffusion¤

In 1D:

\[ \frac{\partial u}{\partial t} = \xi \frac{\partial^4 u}{\partial x^4} \]

In higher dimensions:

\[ \frac{\partial u}{\partial t} = \zeta \nabla \cdot (\nabla \odot \nabla \odot \nabla) u \]

or with spatial mixing:

\[ \frac{\partial u}{\partial t} = \zeta (\nabla \cdot \nabla)(\nabla \cdot \nabla) u \]

exponax.stepper.HyperDiffusion ¤

Bases: BaseStepper

Source code in exponax/stepper/_hyper_diffusion.py 
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class HyperDiffusion(BaseStepper):
    hyper_diffusivity: float
    diffuse_on_diffuse: bool

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        hyper_diffusivity: float = 0.0001,
        diffuse_on_diffuse: bool = False,
    ):
        """
        Timestepper for the d-dimensional (`d ∈ {1, 2, 3}`) hyper-diffusion
        equation on periodic boundary conditions. A hyper-diffusion equation
        acts like a diffusion equation but higher wavenumbers/modes are damped
        even faster.

        In 1d, the hyper-diffusion equation is given by

        ```
            uₜ = - μ uₓₓₓₓ
        ```

        with `μ ∈ ℝ` being the hyper-diffusivity.

        Note the minus sign because by default, a fourth-order derivative
        dampens with a negative coefficient. To match the concept of
        second-order diffusion, a negation is introduced.

        In higher dimensions, the hyper-diffusion equation can be written as

        ```
            uₜ = − μ ((∇⊙∇) ⋅ (∇⊙∇)) u
        ```

        or

        ```
            uₜ = - μ Δ(Δu)
        ```

        The latter introduces spatial mixing.

        **Arguments:**

        - `num_spatial_dims`: The number of spatial dimensions `d`.
        - `domain_extent`: The size of the domain `L`; in higher dimensions
            the domain is assumed to be a scaled hypercube `Ω = (0, L)ᵈ`.
        - `num_points`: The number of points `N` used to discretize the
            domain. This **includes** the left boundary point and **excludes**
            the right boundary point. In higher dimensions; the number of points
            in each dimension is the same. Hence, the total number of degrees of
            freedom is `Nᵈ`.
        - `dt`: The timestep size `Δt` between two consecutive states.
        - `hyper_diffusivity` (keyword-only): The hyper-diffusivity `ν`.
            This stepper only supports scalar (=isotropic) hyper-diffusivity.
            Default: 0.0001.
        - `diffuse_on_diffuse` (keyword-only): If `True`, the second form
            of the hyper-diffusion equation in higher dimensions is used. As a
            consequence, there will be mixing in the spatial derivatives.
            Default: `False`.

        **Notes:**

        - The stepper is unconditionally stable, no matter the choice of
            any argument because the equation is solved analytically in Fourier
            space.
        - Ultimately, only the factor `μ Δt / L⁴` affects the characteristic
            of the dynamics. See also
            [`exponax.stepper.generic.NormalizedLinearStepper`][] with
            `normalized_coefficients = [0, 0, 0, 0, alpha_4]` with `alpha_4 = -
            hyper_diffusivity * dt / domain_extent**4`.
        """
        self.hyper_diffusivity = hyper_diffusivity
        self.diffuse_on_diffuse = diffuse_on_diffuse
        super().__init__(
            num_spatial_dims=num_spatial_dims,
            domain_extent=domain_extent,
            num_points=num_points,
            dt=dt,
            num_channels=1,
            order=0,
        )

    def _build_linear_operator(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> Complex[Array, "1 ... (N//2)+1"]:
        # Use minus sign to have diffusion work in "correct direction" by default
        if self.diffuse_on_diffuse:
            laplace_operator = build_laplace_operator(derivative_operator)
            linear_operator = (
                -self.hyper_diffusivity * laplace_operator * laplace_operator
            )
        else:
            linear_operator = -self.hyper_diffusivity * build_laplace_operator(
                derivative_operator, order=4
            )

        return linear_operator

    def _build_nonlinear_fun(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> ZeroNonlinearFun:
        return ZeroNonlinearFun(
            self.num_spatial_dims,
            self.num_points,
        )
__init__ ¤
__init__(
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    hyper_diffusivity: float = 0.0001,
    diffuse_on_diffuse: bool = False
)

Timestepper for the d-dimensional (d ∈ {1, 2, 3}) hyper-diffusion equation on periodic boundary conditions. A hyper-diffusion equation acts like a diffusion equation but higher wavenumbers/modes are damped even faster.

In 1d, the hyper-diffusion equation is given by

    uₜ = - μ uₓₓₓₓ

with μ ∈ ℝ being the hyper-diffusivity.

Note the minus sign because by default, a fourth-order derivative dampens with a negative coefficient. To match the concept of second-order diffusion, a negation is introduced.

In higher dimensions, the hyper-diffusion equation can be written as

    uₜ = − μ ((∇⊙∇) ⋅ (∇⊙∇)) u

or

    uₜ = - μ Δ(Δu)

The latter introduces spatial mixing.

Arguments:

  • num_spatial_dims: The number of spatial dimensions d.
  • domain_extent: The size of the domain L; in higher dimensions the domain is assumed to be a scaled hypercube Ω = (0, L)ᵈ.
  • num_points: The number of points N used to discretize the domain. This includes the left boundary point and excludes the right boundary point. In higher dimensions; the number of points in each dimension is the same. Hence, the total number of degrees of freedom is Nᵈ.
  • dt: The timestep size Δt between two consecutive states.
  • hyper_diffusivity (keyword-only): The hyper-diffusivity ν. This stepper only supports scalar (=isotropic) hyper-diffusivity. Default: 0.0001.
  • diffuse_on_diffuse (keyword-only): If True, the second form of the hyper-diffusion equation in higher dimensions is used. As a consequence, there will be mixing in the spatial derivatives. Default: False.

Notes:

  • The stepper is unconditionally stable, no matter the choice of any argument because the equation is solved analytically in Fourier space.
  • Ultimately, only the factor μ Δt / L⁴ affects the characteristic of the dynamics. See also exponax.stepper.generic.NormalizedLinearStepper with normalized_coefficients = [0, 0, 0, 0, alpha_4] with alpha_4 = - hyper_diffusivity * dt / domain_extent**4.
Source code in exponax/stepper/_hyper_diffusion.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    hyper_diffusivity: float = 0.0001,
    diffuse_on_diffuse: bool = False,
):
    """
    Timestepper for the d-dimensional (`d ∈ {1, 2, 3}`) hyper-diffusion
    equation on periodic boundary conditions. A hyper-diffusion equation
    acts like a diffusion equation but higher wavenumbers/modes are damped
    even faster.

    In 1d, the hyper-diffusion equation is given by

    ```
        uₜ = - μ uₓₓₓₓ
    ```

    with `μ ∈ ℝ` being the hyper-diffusivity.

    Note the minus sign because by default, a fourth-order derivative
    dampens with a negative coefficient. To match the concept of
    second-order diffusion, a negation is introduced.

    In higher dimensions, the hyper-diffusion equation can be written as

    ```
        uₜ = − μ ((∇⊙∇) ⋅ (∇⊙∇)) u
    ```

    or

    ```
        uₜ = - μ Δ(Δu)
    ```

    The latter introduces spatial mixing.

    **Arguments:**

    - `num_spatial_dims`: The number of spatial dimensions `d`.
    - `domain_extent`: The size of the domain `L`; in higher dimensions
        the domain is assumed to be a scaled hypercube `Ω = (0, L)ᵈ`.
    - `num_points`: The number of points `N` used to discretize the
        domain. This **includes** the left boundary point and **excludes**
        the right boundary point. In higher dimensions; the number of points
        in each dimension is the same. Hence, the total number of degrees of
        freedom is `Nᵈ`.
    - `dt`: The timestep size `Δt` between two consecutive states.
    - `hyper_diffusivity` (keyword-only): The hyper-diffusivity `ν`.
        This stepper only supports scalar (=isotropic) hyper-diffusivity.
        Default: 0.0001.
    - `diffuse_on_diffuse` (keyword-only): If `True`, the second form
        of the hyper-diffusion equation in higher dimensions is used. As a
        consequence, there will be mixing in the spatial derivatives.
        Default: `False`.

    **Notes:**

    - The stepper is unconditionally stable, no matter the choice of
        any argument because the equation is solved analytically in Fourier
        space.
    - Ultimately, only the factor `μ Δt / L⁴` affects the characteristic
        of the dynamics. See also
        [`exponax.stepper.generic.NormalizedLinearStepper`][] with
        `normalized_coefficients = [0, 0, 0, 0, alpha_4]` with `alpha_4 = -
        hyper_diffusivity * dt / domain_extent**4`.
    """
    self.hyper_diffusivity = hyper_diffusivity
    self.diffuse_on_diffuse = diffuse_on_diffuse
    super().__init__(
        num_spatial_dims=num_spatial_dims,
        domain_extent=domain_extent,
        num_points=num_points,
        dt=dt,
        num_channels=1,
        order=0,
    )
__call__ ¤
__call__(
    u: Float[Array, "C ... N"]
) -> Float[Array, "C ... N"]

Perform one step of the time integration for a single state.

Arguments:

  • u: The state vector, shape (C, ..., N,).

Returns:

  • u_next: The state vector after one step, shape (C, ..., N,).

Tip

Use this call method together with exponax.rollout to efficiently produce temporal trajectories.

Info

For batched operation, use jax.vmap on this function.

Source code in exponax/_base_stepper.py 
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def __call__(
    self,
    u: Float[Array, "C ... N"],
) -> Float[Array, "C ... N"]:
    """
    Perform one step of the time integration for a single state.

    **Arguments:**

    - `u`: The state vector, shape `(C, ..., N,)`.

    **Returns:**

    - `u_next`: The state vector after one step, shape `(C, ..., N,)`.

    !!! tip
        Use this call method together with `exponax.rollout` to efficiently
        produce temporal trajectories.

    !!! info
        For batched operation, use `jax.vmap` on this function.
    """
    expected_shape = (self.num_channels,) + spatial_shape(
        self.num_spatial_dims, self.num_points
    )
    if u.shape != expected_shape:
        raise ValueError(
            f"Expected shape {expected_shape}, got {u.shape}. For batched operation use `jax.vmap` on this function."
        )
    return self.step(u)