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Cahn-Hilliad¤

exponax.stepper.reaction.CahnHilliard ¤

Bases: BaseStepper

Source code in exponax/stepper/reaction/_cahn_hilliard.py 
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class CahnHilliard(BaseStepper):
    diffusivity: float
    gamma: 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 = 1e-2,
        gamma: float = 1e-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}`) Cahn-Hilliard
        reaction-diffusion equation on periodic boundary conditions. This model
        is related to the Allen-Cahn equation. In 1d, it reads

        ```
            uₜ = ν ∂ₓₓ (c₃ u³ + c₁ u − γ uₓₓ)
        ```

        with `ν` the diffusivity, `c₁` the first order coefficient, `c₃` the
        third order coefficient, and `γ` the gamma parameter. The state always
        only has one channel. In higher dimensions, the equation reads

        ```
            uₜ = ν Δ (c₃ u³ + c₁ u - γ Δu)
        ```

        Since the Laplace operator is self-multiplied, there will be spatial
        mixing.

        **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`.
        - `gamma`: The gamma parameter `γ`.
        - `first_order_coefficient`: The first order coefficient `c₁`.
        - `third_order_coefficient`: The third order coefficient `c₃`.
        - `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#L89
            for an example IC of the Cahn-Hilliard in 1d.
        """
        self.diffusivity = diffusivity
        self.gamma = gamma
        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 - self.gamma * laplace)
        )
        return linear_operator

    def _build_nonlinear_fun(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> CahnHilliardNonlinearFun:
        return CahnHilliardNonlinearFun(
            self.num_spatial_dims,
            self.num_points,
            derivative_operator=derivative_operator,
            dealiasing_fraction=self.dealiasing_fraction,
            scale=self.diffusivity * self.third_order_coefficient,
        )
__init__ ¤
__init__(
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    diffusivity: float = 0.01,
    gamma: float = 0.001,
    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}) Cahn-Hilliard reaction-diffusion equation on periodic boundary conditions. This model is related to the Allen-Cahn equation. In 1d, it reads

    uₜ = ν ∂ₓₓ (c₃ u³ + c₁ u − γ uₓₓ)

with ν the diffusivity, c₁ the first order coefficient, c₃ the third order coefficient, and γ the gamma parameter. The state always only has one channel. In higher dimensions, the equation reads

    uₜ = ν Δ (c₃ u³ + c₁ u - γ Δu)

Since the Laplace operator is self-multiplied, there will be spatial mixing.

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.
  • gamma: The gamma parameter γ.
  • first_order_coefficient: The first order coefficient c₁.
  • third_order_coefficient: The third order coefficient c₃.
  • 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#L89 for an example IC of the Cahn-Hilliard in 1d.
Source code in exponax/stepper/reaction/_cahn_hilliard.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    diffusivity: float = 1e-2,
    gamma: float = 1e-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}`) Cahn-Hilliard
    reaction-diffusion equation on periodic boundary conditions. This model
    is related to the Allen-Cahn equation. In 1d, it reads

    ```
        uₜ = ν ∂ₓₓ (c₃ u³ + c₁ u − γ uₓₓ)
    ```

    with `ν` the diffusivity, `c₁` the first order coefficient, `c₃` the
    third order coefficient, and `γ` the gamma parameter. The state always
    only has one channel. In higher dimensions, the equation reads

    ```
        uₜ = ν Δ (c₃ u³ + c₁ u - γ Δu)
    ```

    Since the Laplace operator is self-multiplied, there will be spatial
    mixing.

    **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`.
    - `gamma`: The gamma parameter `γ`.
    - `first_order_coefficient`: The first order coefficient `c₁`.
    - `third_order_coefficient`: The third order coefficient `c₃`.
    - `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#L89
        for an example IC of the Cahn-Hilliard in 1d.
    """
    self.diffusivity = diffusivity
    self.gamma = gamma
    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)