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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/_linear.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, not 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.normalized.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, not 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.normalized.NormalizedLinearStepper with normalized_coefficients = [0, 0, alpha_2] with alpha_2 = diffusivity * dt / domain_extent**2.
Source code in exponax/stepper/_linear.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, not 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.normalized.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"]

Performs a check

Source code in exponax/_base_stepper.py 
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def __call__(
    self,
    u: Float[Array, "C ... N"],
) -> Float[Array, "C ... N"]:
    """
    Performs a check
    """
    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)