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Diffusion¤

In 1D:

\[ \frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial x^2} \]

In higher dimensions:

\[ \frac{\partial u}{\partial t} = \nu \nabla \cdot \nabla u \]

or with anisotropic diffusion:

\[ \frac{\partial u}{\partial t} = \nabla \cdot \left( A \nabla u \right) \]

with \(A \in \R^{D \times D}\) symmetric positive definite.

exponax.stepper.Diffusion ¤

Bases: BaseStepper

Source code in exponax/stepper/_diffusion.py 
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class Diffusion(BaseStepper):
    diffusivity: Float[Array, "D D"]

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        diffusivity: Union[
            Float[Array, "D D"],
            Float[Array, "D"],
            float,
        ] = 0.01,
    ):
        """
        Timestepper for the d-dimensional (`d ∈ {1, 2, 3}`) diffusion equation
        on periodic boundary conditions.

        In 1d, the diffusion equation is given by

        ```
            uₜ = ν uₓₓ
        ```

        with `ν ∈ ℝ` being the diffusivity.

        In higher dimensions, the diffusion equation can written using the
        Laplacian operator.

        ```
            uₜ = ν Δu
        ```

        More generally speaking, there can be anistropic diffusivity given by a
        `A ∈ ℝᵈ ˣ ᵈ` sandwiched between the gradient and divergence operators.

        ```
            uₜ = ∇ ⋅ (A ∇u)
        ```

        **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.
        - `diffusivity` (keyword-only): The diffusivity `ν`. In higher
            dimensions, this can be a scalar (=float), a vector of length `d`,
            or a matrix of shape `d ˣ d`. If a scalar is given, the diffusivity
            is assumed to be the same in all spatial dimensions. If a vector (of
            length `d`) is given, the diffusivity varies across dimensions (=>
            diagonal diffusion). For a matrix, there is fully anisotropic
            diffusion. In this case, `A` must be symmetric positive definite
            (SPD). Default: `0.01`.

        **Notes:**

        - The stepper is unconditionally stable, no matter the choice of
            any argument because the equation is solved analytically in Fourier
            space.
        - A `ν > 0` leads to stable and decaying solutions (i.e., energy is
            removed from the system). A `ν < 0` leads to unstable and growing
            solutions (i.e., energy is added to the system).
        - Ultimately, only the factor `ν Δt / L²` affects the characteristic
            of the dynamics. See also
            [`exponax.stepper.generic.NormalizedLinearStepper`][] with
            `normalized_coefficients = [0, 0, alpha_2]` with `alpha_2 =
            diffusivity * dt / domain_extent**2`.
        """
        # ToDo: more sophisticated checks here
        if isinstance(diffusivity, float):
            diffusivity = jnp.diag(jnp.ones(num_spatial_dims)) * diffusivity
        elif len(diffusivity.shape) == 1:
            diffusivity = jnp.diag(diffusivity)
        self.diffusivity = diffusivity
        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"]:
        laplace_outer_producct = (
            derivative_operator[:, None] * derivative_operator[None, :]
        )
        linear_operator = jnp.einsum(
            "ij,ij...->...",
            self.diffusivity,
            laplace_outer_producct,
        )
        # Add the necessary singleton channel axis
        linear_operator = linear_operator[None, ...]
        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,
    *,
    diffusivity: Union[
        Float[Array, "D D"], Float[Array, D], float
    ] = 0.01
)

Timestepper for the d-dimensional (d ∈ {1, 2, 3}) diffusion equation on periodic boundary conditions.

In 1d, the diffusion equation is given by

    uₜ = ν uₓₓ

with ν ∈ ℝ being the diffusivity.

In higher dimensions, the diffusion equation can written using the Laplacian operator.

    uₜ = ν Δu

More generally speaking, there can be anistropic diffusivity given by a A ∈ ℝᵈ ˣ ᵈ sandwiched between the gradient and divergence operators.

    uₜ = ∇ ⋅ (A ∇u)

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.
  • diffusivity (keyword-only): The diffusivity ν. In higher dimensions, this can be a scalar (=float), a vector of length d, or a matrix of shape d ˣ d. If a scalar is given, the diffusivity is assumed to be the same in all spatial dimensions. If a vector (of length d) is given, the diffusivity varies across dimensions (=> diagonal diffusion). For a matrix, there is fully anisotropic diffusion. In this case, A must be symmetric positive definite (SPD). Default: 0.01.

Notes:

  • The stepper is unconditionally stable, no matter the choice of any argument because the equation is solved analytically in Fourier space.
  • A ν > 0 leads to stable and decaying solutions (i.e., energy is removed from the system). A ν < 0 leads to unstable and growing solutions (i.e., energy is added to the system).
  • Ultimately, only the factor ν Δt / L² affects the characteristic of the dynamics. See also exponax.stepper.generic.NormalizedLinearStepper with normalized_coefficients = [0, 0, alpha_2] with alpha_2 = diffusivity * dt / domain_extent**2.
Source code in exponax/stepper/_diffusion.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    diffusivity: Union[
        Float[Array, "D D"],
        Float[Array, "D"],
        float,
    ] = 0.01,
):
    """
    Timestepper for the d-dimensional (`d ∈ {1, 2, 3}`) diffusion equation
    on periodic boundary conditions.

    In 1d, the diffusion equation is given by

    ```
        uₜ = ν uₓₓ
    ```

    with `ν ∈ ℝ` being the diffusivity.

    In higher dimensions, the diffusion equation can written using the
    Laplacian operator.

    ```
        uₜ = ν Δu
    ```

    More generally speaking, there can be anistropic diffusivity given by a
    `A ∈ ℝᵈ ˣ ᵈ` sandwiched between the gradient and divergence operators.

    ```
        uₜ = ∇ ⋅ (A ∇u)
    ```

    **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.
    - `diffusivity` (keyword-only): The diffusivity `ν`. In higher
        dimensions, this can be a scalar (=float), a vector of length `d`,
        or a matrix of shape `d ˣ d`. If a scalar is given, the diffusivity
        is assumed to be the same in all spatial dimensions. If a vector (of
        length `d`) is given, the diffusivity varies across dimensions (=>
        diagonal diffusion). For a matrix, there is fully anisotropic
        diffusion. In this case, `A` must be symmetric positive definite
        (SPD). Default: `0.01`.

    **Notes:**

    - The stepper is unconditionally stable, no matter the choice of
        any argument because the equation is solved analytically in Fourier
        space.
    - A `ν > 0` leads to stable and decaying solutions (i.e., energy is
        removed from the system). A `ν < 0` leads to unstable and growing
        solutions (i.e., energy is added to the system).
    - Ultimately, only the factor `ν Δt / L²` affects the characteristic
        of the dynamics. See also
        [`exponax.stepper.generic.NormalizedLinearStepper`][] with
        `normalized_coefficients = [0, 0, alpha_2]` with `alpha_2 =
        diffusivity * dt / domain_extent**2`.
    """
    # ToDo: more sophisticated checks here
    if isinstance(diffusivity, float):
        diffusivity = jnp.diag(jnp.ones(num_spatial_dims)) * diffusivity
    elif len(diffusivity.shape) == 1:
        diffusivity = jnp.diag(diffusivity)
    self.diffusivity = diffusivity
    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)