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Swift-Hohenberg¤

exponax.stepper.reaction.SwiftHohenberg ¤

Bases: BaseStepper

Source code in exponax/stepper/reaction/_swift_hohenberg.py 
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class SwiftHohenberg(BaseStepper):
    reactivity: float
    critical_number: float
    polynomial_coefficients: tuple[float, ...]
    dealiasing_fraction: float

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        reactivity: float = 0.7,
        critical_number: float = 1.0,
        polynomial_coefficients: tuple[float, ...] = (0.0, 0.0, 1.0, -1.0),
        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}`) Swift-Hohenberg
        reaction-diffusion equation on periodic boundary conditions (works best
        in 2d). This reaction-diffusion equation is a model for pattern
        formation, for example, the fingerprints on a human finger.

        In 1d, the Swift-Hohenberg equation is given by

        ```
            uₜ = r u - (k + ∂ₓₓ)² u + g(u)
        ```

        with `r` the reactivity, `k` the critical number, `∂ₓₓ` the second
        derivative operator. `g(u)` can be any smooth function. This equation
        restricts to the case of polynomial functions, i.e.

        ```
            g(u) = ∑ᵢ cᵢ uⁱ
        ```

        with `cᵢ` the polynomial coefficients.

        The state only has one channel, no matter the spatial dimension. The
        higher dimensional generarlization reads

        ```
            uₜ = r u - (k + Δ)² u + g(u)
        ```

        with `Δ` the Laplacian. Since the Laplacian is squared, there will be
        spatial mixing.

        The expected temporal behavior is a collective pattern formation which
        will be attained in a steady state.

        **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.
        - `reactivity`: The reactivity `r`. Default is `0.7`.
        - `critical_number`: The critical number `k`. Default is `1.0`.
        - `polynomial_coefficients`: The coefficients `cᵢ` of the polynomial
            function `g(u)`. Default is `(0.0, 0.0, 1.0, -1.0)`. This refers to
            a polynomial of `u² - u³`.
        - `dealiasing_fraction`: The fraction of the highest wavenumbers to
            dealias. Default is `1/2` because the default polynomial has a
            highest degree of 3.
        - `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.
        - `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.
        """
        self.reactivity = reactivity
        self.critical_number = critical_number
        self.polynomial_coefficients = polynomial_coefficients
        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=1,
            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, "1 ... (N//2)+1"]:
        laplace = build_laplace_operator(derivative_operator, order=2)
        linear_operator = self.reactivity - (self.critical_number + laplace) ** 2
        return linear_operator

    def _build_nonlinear_fun(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> PolynomialNonlinearFun:
        return PolynomialNonlinearFun(
            self.num_spatial_dims,
            self.num_points,
            dealiasing_fraction=self.dealiasing_fraction,
            coefficients=self.polynomial_coefficients,
        )
__init__ ¤
__init__(
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    reactivity: float = 0.7,
    critical_number: float = 1.0,
    polynomial_coefficients: tuple[float, ...] = (
        0.0,
        0.0,
        1.0,
        -1.0,
    ),
    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}) Swift-Hohenberg reaction-diffusion equation on periodic boundary conditions (works best in 2d). This reaction-diffusion equation is a model for pattern formation, for example, the fingerprints on a human finger.

In 1d, the Swift-Hohenberg equation is given by

    uₜ = r u - (k + ∂ₓₓ)² u + g(u)

with r the reactivity, k the critical number, ∂ₓₓ the second derivative operator. g(u) can be any smooth function. This equation restricts to the case of polynomial functions, i.e.

    g(u) = ∑ᵢ cᵢ uⁱ

with cᵢ the polynomial coefficients.

The state only has one channel, no matter the spatial dimension. The higher dimensional generarlization reads

    uₜ = r u - (k + Δ)² u + g(u)

with Δ the Laplacian. Since the Laplacian is squared, there will be spatial mixing.

The expected temporal behavior is a collective pattern formation which will be attained in a steady state.

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.
  • reactivity: The reactivity r. Default is 0.7.
  • critical_number: The critical number k. Default is 1.0.
  • polynomial_coefficients: The coefficients cᵢ of the polynomial function g(u). Default is (0.0, 0.0, 1.0, -1.0). This refers to a polynomial of u² - u³.
  • dealiasing_fraction: The fraction of the highest wavenumbers to dealias. Default is 1/2 because the default polynomial has a highest degree of 3.
  • 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.
  • 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.
Source code in exponax/stepper/reaction/_swift_hohenberg.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    reactivity: float = 0.7,
    critical_number: float = 1.0,
    polynomial_coefficients: tuple[float, ...] = (0.0, 0.0, 1.0, -1.0),
    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}`) Swift-Hohenberg
    reaction-diffusion equation on periodic boundary conditions (works best
    in 2d). This reaction-diffusion equation is a model for pattern
    formation, for example, the fingerprints on a human finger.

    In 1d, the Swift-Hohenberg equation is given by

    ```
        uₜ = r u - (k + ∂ₓₓ)² u + g(u)
    ```

    with `r` the reactivity, `k` the critical number, `∂ₓₓ` the second
    derivative operator. `g(u)` can be any smooth function. This equation
    restricts to the case of polynomial functions, i.e.

    ```
        g(u) = ∑ᵢ cᵢ uⁱ
    ```

    with `cᵢ` the polynomial coefficients.

    The state only has one channel, no matter the spatial dimension. The
    higher dimensional generarlization reads

    ```
        uₜ = r u - (k + Δ)² u + g(u)
    ```

    with `Δ` the Laplacian. Since the Laplacian is squared, there will be
    spatial mixing.

    The expected temporal behavior is a collective pattern formation which
    will be attained in a steady state.

    **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.
    - `reactivity`: The reactivity `r`. Default is `0.7`.
    - `critical_number`: The critical number `k`. Default is `1.0`.
    - `polynomial_coefficients`: The coefficients `cᵢ` of the polynomial
        function `g(u)`. Default is `(0.0, 0.0, 1.0, -1.0)`. This refers to
        a polynomial of `u² - u³`.
    - `dealiasing_fraction`: The fraction of the highest wavenumbers to
        dealias. Default is `1/2` because the default polynomial has a
        highest degree of 3.
    - `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.
    - `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.
    """
    self.reactivity = reactivity
    self.critical_number = critical_number
    self.polynomial_coefficients = polynomial_coefficients
    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=1,
        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)