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General Convection Stepper¤

exponax.stepper.generic.GeneralConvectionStepper ¤

Bases: BaseStepper

Source code in exponax/stepper/generic/_convection.py 
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class GeneralConvectionStepper(BaseStepper):
    linear_coefficients: tuple[float, ...]
    convection_scale: float
    dealiasing_fraction: float
    single_channel: bool
    conservative: bool

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        linear_coefficients: tuple[float, ...] = (0.0, 0.0, 0.01),
        convection_scale: float = 1.0,
        single_channel: bool = False,
        conservative: bool = False,
        order=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}`) semi-linear PDEs
        consisting of a convection nonlinearity and an arbitrary combination of
        (isotropic) linear derivatives.

        In 1d, the equation is given by

        ```
            uₜ + b₁ 1/2 (u²)ₓ = sum_j a_j uₓˢ

        ```

        with `b₁` the convection coefficient and `a_j` the coefficients of the
        linear operators. `uₓˢ` denotes the s-th derivative of `u` with respect
        to `x`. Oftentimes `b₁ = 1`.

        In the default configuration, the number of channel grows with the
        number of spatial dimensions. The higher dimensional equation reads

        ```
            uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) = sum_j a_j (1⋅∇ʲ)u
        ```

        Alternatively, with `single_channel=True`, the number of channels can be
        kept to constant 1 no matter the number of spatial dimensions.

        Depending on the collection of linear coefficients a range of dynamics
        can be represented, for example:
            - Burgers equation with `a = (0, 0, 0.01)` with `len(a) = 3`
            - KdV equation with `a = (0, 0, 0, 0.01)` with `len(a) = 4`

        **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.
        - `linear_coefficients` (keyword-only): 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, 0.01)` corresponds to the Burgers equation (because of the
            diffusion)
        - `convection_scale` (keyword-only): The scale `b₁` of the
            convection term. Default is `1.0`.
        - `single_channel`: Whether to use the single channel mode in higher
            dimensions. In this case the the convection is `b₁ (∇ ⋅ 1)(u²)`. In
            this case, the state always has a single channel, no matter the
            spatial dimension. Default: False.
        - `conservative`: Whether to use the conservative form of the convection
            term. Default: False.
        - `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.linear_coefficients = linear_coefficients
        self.convection_scale = convection_scale
        self.single_channel = single_channel
        self.dealiasing_fraction = dealiasing_fraction
        self.conservative = conservative

        if single_channel:
            num_channels = 1
        else:
            # number of channels grow with the spatial dimension
            num_channels = num_spatial_dims

        super().__init__(
            num_spatial_dims=num_spatial_dims,
            domain_extent=domain_extent,
            num_points=num_points,
            dt=dt,
            num_channels=num_channels,
            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"]:
        linear_operator = sum(
            jnp.sum(
                c * (derivative_operator) ** i,
                axis=0,
                keepdims=True,
            )
            for i, c in enumerate(self.linear_coefficients)
        )
        return linear_operator

    def _build_nonlinear_fun(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> ConvectionNonlinearFun:
        return ConvectionNonlinearFun(
            self.num_spatial_dims,
            self.num_points,
            derivative_operator=derivative_operator,
            dealiasing_fraction=self.dealiasing_fraction,
            scale=self.convection_scale,
            single_channel=self.single_channel,
            conservative=self.conservative,
        )
__init__ ¤
__init__(
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    linear_coefficients: tuple[float, ...] = (
        0.0,
        0.0,
        0.01,
    ),
    convection_scale: float = 1.0,
    single_channel: bool = False,
    conservative: bool = False,
    order=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}) semi-linear PDEs consisting of a convection nonlinearity and an arbitrary combination of (isotropic) linear derivatives.

In 1d, the equation is given by

    uₜ + b₁ 1/2 (u²)ₓ = sum_j a_j uₓˢ

with b₁ the convection coefficient and a_j the coefficients of the linear operators. uₓˢ denotes the s-th derivative of u with respect to x. Oftentimes b₁ = 1.

In the default configuration, the number of channel grows with the number of spatial dimensions. The higher dimensional equation reads

    uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) = sum_j a_j (1⋅∇ʲ)u

Alternatively, with single_channel=True, the number of channels can be kept to constant 1 no matter the number of spatial dimensions.

Depending on the collection of linear coefficients a range of dynamics can be represented, for example: - Burgers equation with a = (0, 0, 0.01) with len(a) = 3 - KdV equation with a = (0, 0, 0, 0.01) with len(a) = 4

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.
  • linear_coefficients (keyword-only): 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, 0.01) corresponds to the Burgers equation (because of the diffusion)
  • convection_scale (keyword-only): The scale b₁ of the convection term. Default is 1.0.
  • single_channel: Whether to use the single channel mode in higher dimensions. In this case the the convection is b₁ (∇ ⋅ 1)(u²). In this case, the state always has a single channel, no matter the spatial dimension. Default: False.
  • conservative: Whether to use the conservative form of the convection term. Default: False.
  • 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/_convection.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    linear_coefficients: tuple[float, ...] = (0.0, 0.0, 0.01),
    convection_scale: float = 1.0,
    single_channel: bool = False,
    conservative: bool = False,
    order=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}`) semi-linear PDEs
    consisting of a convection nonlinearity and an arbitrary combination of
    (isotropic) linear derivatives.

    In 1d, the equation is given by

    ```
        uₜ + b₁ 1/2 (u²)ₓ = sum_j a_j uₓˢ

    ```

    with `b₁` the convection coefficient and `a_j` the coefficients of the
    linear operators. `uₓˢ` denotes the s-th derivative of `u` with respect
    to `x`. Oftentimes `b₁ = 1`.

    In the default configuration, the number of channel grows with the
    number of spatial dimensions. The higher dimensional equation reads

    ```
        uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) = sum_j a_j (1⋅∇ʲ)u
    ```

    Alternatively, with `single_channel=True`, the number of channels can be
    kept to constant 1 no matter the number of spatial dimensions.

    Depending on the collection of linear coefficients a range of dynamics
    can be represented, for example:
        - Burgers equation with `a = (0, 0, 0.01)` with `len(a) = 3`
        - KdV equation with `a = (0, 0, 0, 0.01)` with `len(a) = 4`

    **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.
    - `linear_coefficients` (keyword-only): 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, 0.01)` corresponds to the Burgers equation (because of the
        diffusion)
    - `convection_scale` (keyword-only): The scale `b₁` of the
        convection term. Default is `1.0`.
    - `single_channel`: Whether to use the single channel mode in higher
        dimensions. In this case the the convection is `b₁ (∇ ⋅ 1)(u²)`. In
        this case, the state always has a single channel, no matter the
        spatial dimension. Default: False.
    - `conservative`: Whether to use the conservative form of the convection
        term. Default: False.
    - `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.linear_coefficients = linear_coefficients
    self.convection_scale = convection_scale
    self.single_channel = single_channel
    self.dealiasing_fraction = dealiasing_fraction
    self.conservative = conservative

    if single_channel:
        num_channels = 1
    else:
        # number of channels grow with the spatial dimension
        num_channels = num_spatial_dims

    super().__init__(
        num_spatial_dims=num_spatial_dims,
        domain_extent=domain_extent,
        num_points=num_points,
        dt=dt,
        num_channels=num_channels,
        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)