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Allen-Cahn¤

exponax.stepper.reaction.AllenCahn ¤

Bases: BaseStepper

Source code in exponax/stepper/reaction/_allen_cahn.py 
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class AllenCahn(BaseStepper):
    diffusivity: float
    first_order_coefficient: float
    third_order_coefficient: float
    dealiasing_fraction: float

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        diffusivity: float = 5e-3,
        first_order_coefficient: float = 1.0,
        third_order_coefficient: float = -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}`) Allen-Cahn
        reaction-diffusion equation on periodic boundary conditions. This
        reaction-diffusion equation is a model for phase separation, for example
        the separation of oil and water.

        In 1d, the Allen-Cahn equation is given by

        ```
            uₜ = ν uₓₓ + c₁ u + c₃ u³
        ```

        with `ν` the diffusivity, `c₁` the first order coefficient, and `c₃` the
        third order coefficient. No matter the spatial dimension, the state
        always only has one channel. In higher dimensions, the equation reads

        ```
            uₜ = ν Δu + c₁ u + c₃ u³
        ```

        with `Δ` the Laplacian.

        The expected temporal behavior is the formation of sharp interfaces
        between the two phases. The limit of the solution is a step function
        that separates the two phases.

        Note that the Allen-Cahn is often solved with Dirichlet boundary
        conditions, but here we use periodic boundary conditions.

        **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`: The diffusivity `ν`. The default value is `5e-3`.
        - `first_order_coefficient`: The first order coefficient `c₁`. The
            default value is `1.0`.
        - `third_order_coefficient`: The third order coefficient `c₃`. The
            default value is `-1.0`.
        - `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.

        **Notes:**

        - See
            https://github.com/chebfun/chebfun/blob/db207bc9f48278ca4def15bf90591bfa44d0801d/spin.m#L48
            for an example IC of the Allen-Cahn in 1d.
        """
        self.diffusivity = diffusivity
        self.first_order_coefficient = first_order_coefficient
        self.third_order_coefficient = third_order_coefficient
        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.diffusivity * laplace + self.first_order_coefficient
        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=[0.0, 0.0, 0.0, self.third_order_coefficient],
        )
__init__ ¤
__init__(
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    diffusivity: float = 0.005,
    first_order_coefficient: float = 1.0,
    third_order_coefficient: float = -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}) Allen-Cahn reaction-diffusion equation on periodic boundary conditions. This reaction-diffusion equation is a model for phase separation, for example the separation of oil and water.

In 1d, the Allen-Cahn equation is given by

    uₜ = ν uₓₓ + c₁ u + c₃ u³

with ν the diffusivity, c₁ the first order coefficient, and c₃ the third order coefficient. No matter the spatial dimension, the state always only has one channel. In higher dimensions, the equation reads

    uₜ = ν Δu + c₁ u + c₃ u³

with Δ the Laplacian.

The expected temporal behavior is the formation of sharp interfaces between the two phases. The limit of the solution is a step function that separates the two phases.

Note that the Allen-Cahn is often solved with Dirichlet boundary conditions, but here we use periodic boundary conditions.

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: The diffusivity ν. The default value is 5e-3.
  • first_order_coefficient: The first order coefficient c₁. The default value is 1.0.
  • third_order_coefficient: The third order coefficient c₃. The default value is -1.0.
  • 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.

Notes:

  • See https://github.com/chebfun/chebfun/blob/db207bc9f48278ca4def15bf90591bfa44d0801d/spin.m#L48 for an example IC of the Allen-Cahn in 1d.
Source code in exponax/stepper/reaction/_allen_cahn.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    diffusivity: float = 5e-3,
    first_order_coefficient: float = 1.0,
    third_order_coefficient: float = -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}`) Allen-Cahn
    reaction-diffusion equation on periodic boundary conditions. This
    reaction-diffusion equation is a model for phase separation, for example
    the separation of oil and water.

    In 1d, the Allen-Cahn equation is given by

    ```
        uₜ = ν uₓₓ + c₁ u + c₃ u³
    ```

    with `ν` the diffusivity, `c₁` the first order coefficient, and `c₃` the
    third order coefficient. No matter the spatial dimension, the state
    always only has one channel. In higher dimensions, the equation reads

    ```
        uₜ = ν Δu + c₁ u + c₃ u³
    ```

    with `Δ` the Laplacian.

    The expected temporal behavior is the formation of sharp interfaces
    between the two phases. The limit of the solution is a step function
    that separates the two phases.

    Note that the Allen-Cahn is often solved with Dirichlet boundary
    conditions, but here we use periodic boundary conditions.

    **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`: The diffusivity `ν`. The default value is `5e-3`.
    - `first_order_coefficient`: The first order coefficient `c₁`. The
        default value is `1.0`.
    - `third_order_coefficient`: The third order coefficient `c₃`. The
        default value is `-1.0`.
    - `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.

    **Notes:**

    - See
        https://github.com/chebfun/chebfun/blob/db207bc9f48278ca4def15bf90591bfa44d0801d/spin.m#L48
        for an example IC of the Allen-Cahn in 1d.
    """
    self.diffusivity = diffusivity
    self.first_order_coefficient = first_order_coefficient
    self.third_order_coefficient = third_order_coefficient
    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)