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Gray-Scott¤

exponax.stepper.reaction.GrayScott ¤

Bases: BaseStepper

Source code in exponax/stepper/reaction/_gray_scott.py
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class GrayScott(BaseStepper):
    diffusivity_1: float
    diffusivity_2: float
    feed_rate: float
    kill_rate: float
    dealiasing_fraction: float

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        diffusivity_1: float = 2e-5,
        diffusivity_2: float = 1e-5,
        feed_rate: float = 0.04,
        kill_rate: float = 0.06,
        order: int = 2,
        # Needs lower value due to cubic nonlinearity
        dealiasing_fraction: float = 1 / 2,
        num_circle_points: int = 16,
        circle_radius: float = 1.0,
    ):
        """
        Timestepper for the d-dimensional (`d ∈ {1, 2, 3}`) Gray-Scott reaction
        diffusion equation on periodic boundary conditions. This
        reaction-diffusion models the interaction of two chemical species u & v.

        In 1d, the Gray-Scott equation is given by

        ```
            uₜ = ν₁ uₓₓ + f(1 - u) - u v²

            vₜ = ν₂ vₓₓ - (f + k) v + u v²
        ```

        with `ν₁` and `ν₂` the diffusivities, `f` the feed rate, and `k` the
        kill rate. No matter the spatial dimension, this dynamics always has two
        channels, refering to the two chemical species. In higher dimensions,
        the equations read

        ```
            uₜ = ν₁ Δu + f(1 - u) - u v²

            vₜ = ν₂ Δv - (f + k) v + u v²
        ```

        with `Δ` the Laplacian.

        The Gray-Scott equation is known to produce a variety of patterns, such
        as spots, stripes, and spirals. The expected temporal behavior is highly
        dependent on the values of the feed and kill rates, see also this paper:
        https://www.ljll.fr/hecht/ftp/ff++/2015-cimpa-IIT/edp-tuto/Pearson.pdf

        IMPORTANT: Both channels are expected to have values in the range `[0,
        1]`.

        **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_1`: The diffusivity `ν₁` of the first species.
            Default is `2e-5`.
        - `diffusivity_2`: The diffusivity `ν₂` of the second species.
            Default is `1e-5`.
        - `feed_rate`: The feed rate `f`. Default is `0.04`.
        - `kill_rate`: The kill rate `k`. Default is `0.06`.
        - `order`: The order of the Exponential Time Differencing Runge
            Kutta method. Must be one of {0, 1, 2, 3, 4}. The option `0` only
            solves the linear part of the equation. Use higher values for higher
            accuracy and stability. The default choice of `2` is a good
            compromise for single precision floats.
        - `dealiasing_fraction`: The fraction of the wavenumbers to keep
            before evaluating the nonlinearity. Default: 1/2.
        - `num_circle_points`: How many points to use in the complex contour
            integral method to compute the coefficients of the exponential time
            differencing Runge Kutta method. Default: 16.
        - `circle_radius`: The radius of the contour used to compute the
            coefficients of the exponential time differencing Runge Kutta
            method. Default: 1.0.

        TODO: Translate the different configurations of
        https://www.ljll.fr/hecht/ftp/ff++/2015-cimpa-IIT/edp-tuto/Pearson.pdf
        """
        self.diffusivity_1 = diffusivity_1
        self.diffusivity_2 = diffusivity_2
        self.feed_rate = feed_rate
        self.kill_rate = kill_rate
        self.dealiasing_fraction = dealiasing_fraction
        super().__init__(
            num_spatial_dims=num_spatial_dims,
            domain_extent=domain_extent,
            num_points=num_points,
            dt=dt,
            num_channels=2,
            order=order,
            num_circle_points=num_circle_points,
            circle_radius=circle_radius,
        )

    def _build_linear_operator(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> Complex[Array, "2 ... (N//2)+1"]:
        laplace = build_laplace_operator(derivative_operator, order=2)
        linear_operator = jnp.concatenate(
            [
                self.diffusivity_1 * laplace,
                self.diffusivity_2 * laplace,
            ]
        )
        return linear_operator

    def _build_nonlinear_fun(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> GrayScottNonlinearFun:
        return GrayScottNonlinearFun(
            self.num_spatial_dims,
            self.num_points,
            feed_rate=self.feed_rate,
            kill_rate=self.kill_rate,
            dealiasing_fraction=self.dealiasing_fraction,
        )
__init__ ¤
__init__(
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    diffusivity_1: float = 2e-05,
    diffusivity_2: float = 1e-05,
    feed_rate: float = 0.04,
    kill_rate: float = 0.06,
    order: int = 2,
    dealiasing_fraction: float = 1 / 2,
    num_circle_points: int = 16,
    circle_radius: float = 1.0
)

Timestepper for the d-dimensional (d ∈ {1, 2, 3}) Gray-Scott reaction diffusion equation on periodic boundary conditions. This reaction-diffusion models the interaction of two chemical species u & v.

In 1d, the Gray-Scott equation is given by

    uₜ = ν₁ uₓₓ + f(1 - u) - u v²

    vₜ = ν₂ vₓₓ - (f + k) v + u v²

with ν₁ and ν₂ the diffusivities, f the feed rate, and k the kill rate. No matter the spatial dimension, this dynamics always has two channels, refering to the two chemical species. In higher dimensions, the equations read

    uₜ = ν₁ Δu + f(1 - u) - u v²

    vₜ = ν₂ Δv - (f + k) v + u v²

with Δ the Laplacian.

The Gray-Scott equation is known to produce a variety of patterns, such as spots, stripes, and spirals. The expected temporal behavior is highly dependent on the values of the feed and kill rates, see also this paper: https://www.ljll.fr/hecht/ftp/ff++/2015-cimpa-IIT/edp-tuto/Pearson.pdf

IMPORTANT: Both channels are expected to have values in the range [0, 1].

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_1: The diffusivity ν₁ of the first species. Default is 2e-5.
  • diffusivity_2: The diffusivity ν₂ of the second species. Default is 1e-5.
  • feed_rate: The feed rate f. Default is 0.04.
  • kill_rate: The kill rate k. Default is 0.06.
  • order: The order of the Exponential Time Differencing Runge Kutta method. Must be one of {0, 1, 2, 3, 4}. The option 0 only solves the linear part of the equation. Use higher values for higher accuracy and stability. The default choice of 2 is a good compromise for single precision floats.
  • dealiasing_fraction: The fraction of the wavenumbers to keep before evaluating the nonlinearity. Default: 1/2.
  • num_circle_points: How many points to use in the complex contour integral method to compute the coefficients of the exponential time differencing Runge Kutta method. Default: 16.
  • circle_radius: The radius of the contour used to compute the coefficients of the exponential time differencing Runge Kutta method. Default: 1.0.

TODO: Translate the different configurations of https://www.ljll.fr/hecht/ftp/ff++/2015-cimpa-IIT/edp-tuto/Pearson.pdf

Source code in exponax/stepper/reaction/_gray_scott.py
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    diffusivity_1: float = 2e-5,
    diffusivity_2: float = 1e-5,
    feed_rate: float = 0.04,
    kill_rate: float = 0.06,
    order: int = 2,
    # Needs lower value due to cubic nonlinearity
    dealiasing_fraction: float = 1 / 2,
    num_circle_points: int = 16,
    circle_radius: float = 1.0,
):
    """
    Timestepper for the d-dimensional (`d ∈ {1, 2, 3}`) Gray-Scott reaction
    diffusion equation on periodic boundary conditions. This
    reaction-diffusion models the interaction of two chemical species u & v.

    In 1d, the Gray-Scott equation is given by

    ```
        uₜ = ν₁ uₓₓ + f(1 - u) - u v²

        vₜ = ν₂ vₓₓ - (f + k) v + u v²
    ```

    with `ν₁` and `ν₂` the diffusivities, `f` the feed rate, and `k` the
    kill rate. No matter the spatial dimension, this dynamics always has two
    channels, refering to the two chemical species. In higher dimensions,
    the equations read

    ```
        uₜ = ν₁ Δu + f(1 - u) - u v²

        vₜ = ν₂ Δv - (f + k) v + u v²
    ```

    with `Δ` the Laplacian.

    The Gray-Scott equation is known to produce a variety of patterns, such
    as spots, stripes, and spirals. The expected temporal behavior is highly
    dependent on the values of the feed and kill rates, see also this paper:
    https://www.ljll.fr/hecht/ftp/ff++/2015-cimpa-IIT/edp-tuto/Pearson.pdf

    IMPORTANT: Both channels are expected to have values in the range `[0,
    1]`.

    **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_1`: The diffusivity `ν₁` of the first species.
        Default is `2e-5`.
    - `diffusivity_2`: The diffusivity `ν₂` of the second species.
        Default is `1e-5`.
    - `feed_rate`: The feed rate `f`. Default is `0.04`.
    - `kill_rate`: The kill rate `k`. Default is `0.06`.
    - `order`: The order of the Exponential Time Differencing Runge
        Kutta method. Must be one of {0, 1, 2, 3, 4}. The option `0` only
        solves the linear part of the equation. Use higher values for higher
        accuracy and stability. The default choice of `2` is a good
        compromise for single precision floats.
    - `dealiasing_fraction`: The fraction of the wavenumbers to keep
        before evaluating the nonlinearity. Default: 1/2.
    - `num_circle_points`: How many points to use in the complex contour
        integral method to compute the coefficients of the exponential time
        differencing Runge Kutta method. Default: 16.
    - `circle_radius`: The radius of the contour used to compute the
        coefficients of the exponential time differencing Runge Kutta
        method. Default: 1.0.

    TODO: Translate the different configurations of
    https://www.ljll.fr/hecht/ftp/ff++/2015-cimpa-IIT/edp-tuto/Pearson.pdf
    """
    self.diffusivity_1 = diffusivity_1
    self.diffusivity_2 = diffusivity_2
    self.feed_rate = feed_rate
    self.kill_rate = kill_rate
    self.dealiasing_fraction = dealiasing_fraction
    super().__init__(
        num_spatial_dims=num_spatial_dims,
        domain_extent=domain_extent,
        num_points=num_points,
        dt=dt,
        num_channels=2,
        order=order,
        num_circle_points=num_circle_points,
        circle_radius=circle_radius,
    )
__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)