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Gradient NormA¤

exponax.stepper.generic.NormalizedGradientNormStepper ¤

Bases: GeneralGradientNormStepper

Source code in exponax/stepper/generic/_gradient_norm.py 
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class NormalizedGradientNormStepper(GeneralGradientNormStepper):
    normalized_linear_coefficients: tuple[float, ...]
    normalized_gradient_norm_scale: float

    def __init__(
        self,
        num_spatial_dims: int,
        num_points: int,
        *,
        normalized_linear_coefficients: tuple[float, ...] = (
            0.0,
            0.0,
            -1.0 * 0.1 / (60.0**2),
            0.0,
            -1.0 * 0.1 / (60.0**4),
        ),
        normalized_gradient_norm_scale: float = 1.0 * 0.1 / (60.0**2),
        order: int = 2,
        dealiasing_fraction: float = 2 / 3,
        num_circle_points: int = 16,
        circle_radius: float = 1.0,
    ):
        """
        Timestepper for the **normalized** d-dimensional (`d ∈ {1, 2, 3}`)
        semi-linear PDEs consisting of a gradient norm nonlinearity and an
        arbitrary combination of (isotropic) linear operators. Uses a normalized
        interface, i.e., the domain is scaled to `Ω = (0, 1)ᵈ` and time step
        size is `Δt = 1.0`.

        See `exponax.stepper.generic.GeneralGradientNormStepper` for more
        details on the functional form of the PDE.

        The number of channels do **not** grow with the number of spatial
        dimensions. They are always one.

        Under the default settings, it behaves like the Kuramoto-Sivashinsky
        equation in combustion format under the following settings.

        By default: the KS equation on L=60.0

        ```python

        exponax.stepper.KuramotoSivashinsky(
            num_spatial_dims=D, domain_extent=60.0, num_points=N, dt=0.1,
            gradient_norm_scale=1.0, second_order_diffusivity=1.0,
            fourth_order_diffusivity=1.0,
        )
        ```

        Note that the coefficient list requires a negative sign because the
        linear derivatives are moved to the right-hand side in this generic
        interface.

        **Arguments:**

        - `num_spatial_dims`: The number of spatial dimensions `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`.
        - `normalized_coefficients`: The list of coefficients `a_j`
            corresponding to the derivatives. The length of this tuple
            represents the highest occuring derivative. The default value `(0.0,
            0.0, -1.0 * 0.1 / (60.0**2), 0.0, -1.0 * 0.1 / (60.0**4))`
            corresponds to the Kuramoto-Sivashinsky equation in combustion
            format on a domain of size `L=60.0` with a time step size of
            `Δt=0.1`.
        - `normalized_gradient_norm_scale`: The scale of the gradient
            norm term `b₂`. Default: `1.0 * 0.1 / (60.0**2)`.
        - `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.
        """
        self.normalized_linear_coefficients = normalized_linear_coefficients
        self.normalized_gradient_norm_scale = normalized_gradient_norm_scale
        super().__init__(
            num_spatial_dims=num_spatial_dims,
            domain_extent=1.0,
            num_points=num_points,
            dt=1.0,
            linear_coefficients=normalized_linear_coefficients,
            gradient_norm_scale=normalized_gradient_norm_scale,
            order=order,
            dealiasing_fraction=dealiasing_fraction,
            num_circle_points=num_circle_points,
            circle_radius=circle_radius,
        )
__init__ ¤
__init__(
    num_spatial_dims: int,
    num_points: int,
    *,
    normalized_linear_coefficients: tuple[float, ...] = (
        0.0,
        0.0,
        -1.0 * 0.1 / 60.0**2,
        0.0,
        -1.0 * 0.1 / 60.0**4,
    ),
    normalized_gradient_norm_scale: float = 1.0
    * 0.1
    / 60.0**2,
    order: int = 2,
    dealiasing_fraction: float = 2 / 3,
    num_circle_points: int = 16,
    circle_radius: float = 1.0
)

Timestepper for the normalized d-dimensional (d ∈ {1, 2, 3}) semi-linear PDEs consisting of a gradient norm nonlinearity and an arbitrary combination of (isotropic) linear operators. Uses a normalized interface, i.e., the domain is scaled to Ω = (0, 1)ᵈ and time step size is Δt = 1.0.

See exponax.stepper.generic.GeneralGradientNormStepper for more details on the functional form of the PDE.

The number of channels do not grow with the number of spatial dimensions. They are always one.

Under the default settings, it behaves like the Kuramoto-Sivashinsky equation in combustion format under the following settings.

By default: the KS equation on L=60.0

exponax.stepper.KuramotoSivashinsky(
    num_spatial_dims=D, domain_extent=60.0, num_points=N, dt=0.1,
    gradient_norm_scale=1.0, second_order_diffusivity=1.0,
    fourth_order_diffusivity=1.0,
)

Note that the coefficient list requires a negative sign because the linear derivatives are moved to the right-hand side in this generic interface.

Arguments:

  • num_spatial_dims: The number of spatial dimensions 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.
  • normalized_coefficients: The list of coefficients a_j corresponding to the derivatives. The length of this tuple represents the highest occuring derivative. The default value (0.0, 0.0, -1.0 * 0.1 / (60.0**2), 0.0, -1.0 * 0.1 / (60.0**4)) corresponds to the Kuramoto-Sivashinsky equation in combustion format on a domain of size L=60.0 with a time step size of Δt=0.1.
  • normalized_gradient_norm_scale: The scale of the gradient norm term b₂. Default: 1.0 * 0.1 / (60.0**2).
  • 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.
Source code in exponax/stepper/generic/_gradient_norm.py 
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def __init__(
    self,
    num_spatial_dims: int,
    num_points: int,
    *,
    normalized_linear_coefficients: tuple[float, ...] = (
        0.0,
        0.0,
        -1.0 * 0.1 / (60.0**2),
        0.0,
        -1.0 * 0.1 / (60.0**4),
    ),
    normalized_gradient_norm_scale: float = 1.0 * 0.1 / (60.0**2),
    order: int = 2,
    dealiasing_fraction: float = 2 / 3,
    num_circle_points: int = 16,
    circle_radius: float = 1.0,
):
    """
    Timestepper for the **normalized** d-dimensional (`d ∈ {1, 2, 3}`)
    semi-linear PDEs consisting of a gradient norm nonlinearity and an
    arbitrary combination of (isotropic) linear operators. Uses a normalized
    interface, i.e., the domain is scaled to `Ω = (0, 1)ᵈ` and time step
    size is `Δt = 1.0`.

    See `exponax.stepper.generic.GeneralGradientNormStepper` for more
    details on the functional form of the PDE.

    The number of channels do **not** grow with the number of spatial
    dimensions. They are always one.

    Under the default settings, it behaves like the Kuramoto-Sivashinsky
    equation in combustion format under the following settings.

    By default: the KS equation on L=60.0

    ```python

    exponax.stepper.KuramotoSivashinsky(
        num_spatial_dims=D, domain_extent=60.0, num_points=N, dt=0.1,
        gradient_norm_scale=1.0, second_order_diffusivity=1.0,
        fourth_order_diffusivity=1.0,
    )
    ```

    Note that the coefficient list requires a negative sign because the
    linear derivatives are moved to the right-hand side in this generic
    interface.

    **Arguments:**

    - `num_spatial_dims`: The number of spatial dimensions `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`.
    - `normalized_coefficients`: The list of coefficients `a_j`
        corresponding to the derivatives. The length of this tuple
        represents the highest occuring derivative. The default value `(0.0,
        0.0, -1.0 * 0.1 / (60.0**2), 0.0, -1.0 * 0.1 / (60.0**4))`
        corresponds to the Kuramoto-Sivashinsky equation in combustion
        format on a domain of size `L=60.0` with a time step size of
        `Δt=0.1`.
    - `normalized_gradient_norm_scale`: The scale of the gradient
        norm term `b₂`. Default: `1.0 * 0.1 / (60.0**2)`.
    - `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.
    """
    self.normalized_linear_coefficients = normalized_linear_coefficients
    self.normalized_gradient_norm_scale = normalized_gradient_norm_scale
    super().__init__(
        num_spatial_dims=num_spatial_dims,
        domain_extent=1.0,
        num_points=num_points,
        dt=1.0,
        linear_coefficients=normalized_linear_coefficients,
        gradient_norm_scale=normalized_gradient_norm_scale,
        order=order,
        dealiasing_fraction=dealiasing_fraction,
        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)