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Fisher-KPP¤

exponax.stepper.reaction.FisherKPP ¤

Bases: BaseStepper

Source code in exponax/stepper/reaction/_fisher_kpp.py 
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class FisherKPP(BaseStepper):
    diffusivity: float
    reactivity: float
    dealiasing_fraction: float

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        diffusivity: float = 0.01,
        reactivity=1.0,
        order: int = 2,
        dealiasing_fraction: float = 2 / 3,
        num_circle_points: int = 16,
        circle_radius: float = 1.0,
    ):
        """
        Timestepper for the d-dimensional (`d ∈ {1, 2, 3}`) Fisher-KPP equation
        on periodic boundary conditions. This reaction-diffusion equation is
        related to logistic growth and describes the spread of a population.

        In 1d, the Fisher-KPP equation is given by

        ```
            uₜ = ν uₓₓ + r u (1 - u)
        ```

        with `ν` the diffusivity and `r` the reactivity. In 1d, the state `u`
        has only one channel. As such the discretized state is represented by a
        tensor of shape `(1, num_points)`. For higher dimensions, the number of
        channels will be constant 1, no matter the dimension. The
        higher-dimensional equation reads

        ```
            uₜ = ν Δu + r u (1 - u)
        ```

        with `Δ` the Laplacian.

        The dynamics requires initial conditions in the range `[0, 1]`. Then,
        the expected temporal behavior is a collective spread and growth. The
        limit of the solution is the constant state `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`: The diffusivity `ν`.
        - `reactivity`: The reactivity `r`.
        - `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. The default 2/3 corresponds to
            Orszag's 2/3 rule. To fully eliminate aliasing, use 1/2. Default:
            2/3.
        - `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:**

        - The dynamics require initial conditions in the range `[0, 1]`.
            This can be achieved by combining any of the available IC generators
            with the [`exponax.ic.ClampingICGenerator`]. Alternatively, a good
            choice is also the [`exponax.ic.GaussianBlobs`]

        **Good Values:**

        - Use the `ClampingICGenerator` on `RandomTruncatedFourierSeries`
            with limits `[0, 1]` to generate initial conditions. Set
            `domain_extent = 1.0`, `num_points = 100`, `dt = 0.001`, and produce
            a trajectory of 500 steps. The final state of almost constant `1`
            will be reached after 200-400 steps.
        """
        self.dealiasing_fraction = dealiasing_fraction
        self.diffusivity = diffusivity
        self.reactivity = reactivity
        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.reactivity
        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, -self.reactivity],
        )
__init__ ¤
__init__(
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    diffusivity: float = 0.01,
    reactivity=1.0,
    order: int = 2,
    dealiasing_fraction: float = 2 / 3,
    num_circle_points: int = 16,
    circle_radius: float = 1.0
)

Timestepper for the d-dimensional (d ∈ {1, 2, 3}) Fisher-KPP equation on periodic boundary conditions. This reaction-diffusion equation is related to logistic growth and describes the spread of a population.

In 1d, the Fisher-KPP equation is given by

    uₜ = ν uₓₓ + r u (1 - u)

with ν the diffusivity and r the reactivity. In 1d, the state u has only one channel. As such the discretized state is represented by a tensor of shape (1, num_points). For higher dimensions, the number of channels will be constant 1, no matter the dimension. The higher-dimensional equation reads

    uₜ = ν Δu + r u (1 - u)

with Δ the Laplacian.

The dynamics requires initial conditions in the range [0, 1]. Then, the expected temporal behavior is a collective spread and growth. The limit of the solution is the constant state 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: The diffusivity ν.
  • reactivity: The reactivity r.
  • 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. The default 2/3 corresponds to Orszag's 2/3 rule. To fully eliminate aliasing, use 1/2. Default: 2/3.
  • 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:

  • The dynamics require initial conditions in the range [0, 1]. This can be achieved by combining any of the available IC generators with the [exponax.ic.ClampingICGenerator]. Alternatively, a good choice is also the [exponax.ic.GaussianBlobs]

Good Values:

  • Use the ClampingICGenerator on RandomTruncatedFourierSeries with limits [0, 1] to generate initial conditions. Set domain_extent = 1.0, num_points = 100, dt = 0.001, and produce a trajectory of 500 steps. The final state of almost constant 1 will be reached after 200-400 steps.
Source code in exponax/stepper/reaction/_fisher_kpp.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    diffusivity: float = 0.01,
    reactivity=1.0,
    order: int = 2,
    dealiasing_fraction: float = 2 / 3,
    num_circle_points: int = 16,
    circle_radius: float = 1.0,
):
    """
    Timestepper for the d-dimensional (`d ∈ {1, 2, 3}`) Fisher-KPP equation
    on periodic boundary conditions. This reaction-diffusion equation is
    related to logistic growth and describes the spread of a population.

    In 1d, the Fisher-KPP equation is given by

    ```
        uₜ = ν uₓₓ + r u (1 - u)
    ```

    with `ν` the diffusivity and `r` the reactivity. In 1d, the state `u`
    has only one channel. As such the discretized state is represented by a
    tensor of shape `(1, num_points)`. For higher dimensions, the number of
    channels will be constant 1, no matter the dimension. The
    higher-dimensional equation reads

    ```
        uₜ = ν Δu + r u (1 - u)
    ```

    with `Δ` the Laplacian.

    The dynamics requires initial conditions in the range `[0, 1]`. Then,
    the expected temporal behavior is a collective spread and growth. The
    limit of the solution is the constant state `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`: The diffusivity `ν`.
    - `reactivity`: The reactivity `r`.
    - `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. The default 2/3 corresponds to
        Orszag's 2/3 rule. To fully eliminate aliasing, use 1/2. Default:
        2/3.
    - `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:**

    - The dynamics require initial conditions in the range `[0, 1]`.
        This can be achieved by combining any of the available IC generators
        with the [`exponax.ic.ClampingICGenerator`]. Alternatively, a good
        choice is also the [`exponax.ic.GaussianBlobs`]

    **Good Values:**

    - Use the `ClampingICGenerator` on `RandomTruncatedFourierSeries`
        with limits `[0, 1]` to generate initial conditions. Set
        `domain_extent = 1.0`, `num_points = 100`, `dt = 0.001`, and produce
        a trajectory of 500 steps. The final state of almost constant `1`
        will be reached after 200-400 steps.
    """
    self.dealiasing_fraction = dealiasing_fraction
    self.diffusivity = diffusivity
    self.reactivity = reactivity
    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)