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Korteweg-de Vries¤

exponax.stepper.KortewegDeVries ¤

Bases: BaseStepper

Source code in exponax/stepper/_korteweg_de_vries.py 
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class KortewegDeVries(BaseStepper):
    convection_scale: float
    dispersivity: float
    diffusivity: float
    hyper_diffusivity: float
    dealiasing_fraction: float
    advect_over_diffuse: bool
    diffuse_over_diffuse: bool
    single_channel: bool
    conservative: bool

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        convection_scale: float = -6.0,
        diffusivity: float = 0.0,
        dispersivity: float = 1.0,
        hyper_diffusivity: float = 0.01,
        advect_over_diffuse: bool = False,
        diffuse_over_diffuse: bool = False,
        single_channel: bool = False,
        conservative: bool = False,
        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}`) (hyper-viscous)
        Korteweg-de Vries equation on periodic boundary conditions.

        In 1d, the Korteweg-de Vries equation is given by

        ```
            uₜ + b₁ 1/2 (u²)ₓ + a₃ uₓₓₓ = ν uₓₓ - μ uₓₓₓₓ
        ```

        with `b₁` the convection coefficient, `a₃` the dispersion coefficient
        and `ν` the diffusivity. Oftentimes `b₁ = -6` and `ν = 0`. The
        nonlinearity is similar to the Burgers equation and the number of
        channels grow with the number of spatial dimensions. In higher
        dimensions, the equation reads (using vector format for the channels)

        ```
            uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) + a₃ 1 ⋅ (∇⊙∇⊙(∇u)) = ν Δu - μ ((∇⊙∇) ⋅ (∇⊙∇)) u
        ```

        or

        ```
            uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) + a₃ ∇ ⋅ ∇(Δu) = ν Δu - μ Δ(Δu)
        ```

        if `advect_over_diffuse` is `True` and `diffuse_on_diffuse` is `True`,

        In 1d, the expected temporal behavior is that the initial condition
        breaks into soliton waves that propagate at a speed depending on their
        height. They interact with other soliton waves by being spatially
        displaced but having an unchanged shape and propagation speed. If either
        the `diffusivity` or the `hyper_diffusivity` is non-zero (by default,
        the latter is active), the solution will decay over time with the
        produced small-scall features decaying the fastest.

        **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.
        - `convection_scale`: The convection coefficient `b₁`. Note that the
            convection is already scaled by 1/2 to account for the conservative
            evaluation. The value of `b₁` scales it further. Oftentimes `b₁ =
            -6` to match the analytical soliton solutions. See also
            https://en.wikipedia.org/wiki/Korteweg%E2%80%93De_Vries_equation#One-soliton_solution
        - `diffusivity`: The rate at which the solution decays.
        - `dispersivity`: The dispersion coefficient `a₃`. Dispersion refers
            to wavenumber-dependent advection, i.e., higher wavenumbers are
            advected faster. Default `1.0`,
        - `hyper_diffusivity`: The hyper-diffusivity (associated with a
            fourth-order term). This stepper only supports scalar (=isotropic)
            hyper-diffusivity. Default: 0.01.
        - `advect_over_diffuse`: If `True`, the dispersion is computed as
            advection over diffusion. This adds spatial mixing. Default is
            `False`.
        - `diffuse_on_diffuse`: If `True`, the hyper-diffusion is computed as
            the diffusion of the diffusion. This adds spatial mixing. Default is
            `False`.
        - `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.

        **Notes:**

            -

        **Good Values:**

        - There is an anlytical solution to the (inviscid, `ν = 0`) KdV of
            `u(t, x) = - 1/2 c^2 sech^2(c/2 (x - ct - a))` with the hyperbolic
            secant `sech` and arbitrarily selected speed `c` and shift `a`.
        - For a nice simulation with an initial condition that breaks into
            solitons choose `domain_extent=20.0` and an initial condition with
            the first 5-10 modes. Set dt=0.01, num points in the range of 50-200
            are sufficient.
        """
        self.convection_scale = convection_scale
        self.diffusivity = diffusivity
        self.dispersivity = dispersivity
        self.hyper_diffusivity = hyper_diffusivity
        self.advect_over_diffuse = advect_over_diffuse
        self.diffuse_over_diffuse = diffuse_over_diffuse
        self.single_channel = single_channel
        self.conservative = conservative
        self.dealiasing_fraction = dealiasing_fraction

        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"]:
        dispersion_velocity = self.dispersivity * jnp.ones(self.num_spatial_dims)
        laplace_operator = build_laplace_operator(derivative_operator, order=2)

        diffusion_operator = self.diffusivity * laplace_operator

        if self.advect_over_diffuse:
            dispersion_operator = (
                -build_gradient_inner_product_operator(
                    derivative_operator, self.advect_over_diffuse_dispersivity, order=1
                )
                * laplace_operator
            )
        else:
            dispersion_operator = -build_gradient_inner_product_operator(
                derivative_operator, dispersion_velocity, order=3
            )

        if self.diffuse_over_diffuse:
            hyper_diffusion_operator = (
                -self.hyper_diffusivity * laplace_operator * laplace_operator
            )
        else:
            hyper_diffusion_operator = -self.hyper_diffusivity * build_laplace_operator(
                derivative_operator, order=4
            )

        linear_operator = (
            diffusion_operator + dispersion_operator + hyper_diffusion_operator
        )
        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,
    *,
    convection_scale: float = -6.0,
    diffusivity: float = 0.0,
    dispersivity: float = 1.0,
    hyper_diffusivity: float = 0.01,
    advect_over_diffuse: bool = False,
    diffuse_over_diffuse: bool = False,
    single_channel: bool = False,
    conservative: bool = False,
    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}) (hyper-viscous) Korteweg-de Vries equation on periodic boundary conditions.

In 1d, the Korteweg-de Vries equation is given by

    uₜ + b₁ 1/2 (u²)ₓ + a₃ uₓₓₓ = ν uₓₓ - μ uₓₓₓₓ

with b₁ the convection coefficient, a₃ the dispersion coefficient and ν the diffusivity. Oftentimes b₁ = -6 and ν = 0. The nonlinearity is similar to the Burgers equation and the number of channels grow with the number of spatial dimensions. In higher dimensions, the equation reads (using vector format for the channels)

    uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) + a₃ 1 ⋅ (∇⊙∇⊙(∇u)) = ν Δu - μ ((∇⊙∇) ⋅ (∇⊙∇)) u

or

    uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) + a₃ ∇ ⋅ ∇(Δu) = ν Δu - μ Δ(Δu)

if advect_over_diffuse is True and diffuse_on_diffuse is True,

In 1d, the expected temporal behavior is that the initial condition breaks into soliton waves that propagate at a speed depending on their height. They interact with other soliton waves by being spatially displaced but having an unchanged shape and propagation speed. If either the diffusivity or the hyper_diffusivity is non-zero (by default, the latter is active), the solution will decay over time with the produced small-scall features decaying the fastest.

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.
  • convection_scale: The convection coefficient b₁. Note that the convection is already scaled by 1/2 to account for the conservative evaluation. The value of b₁ scales it further. Oftentimes b₁ = -6 to match the analytical soliton solutions. See also https://en.wikipedia.org/wiki/Korteweg%E2%80%93De_Vries_equation#One-soliton_solution
  • diffusivity: The rate at which the solution decays.
  • dispersivity: The dispersion coefficient a₃. Dispersion refers to wavenumber-dependent advection, i.e., higher wavenumbers are advected faster. Default 1.0,
  • hyper_diffusivity: The hyper-diffusivity (associated with a fourth-order term). This stepper only supports scalar (=isotropic) hyper-diffusivity. Default: 0.01.
  • advect_over_diffuse: If True, the dispersion is computed as advection over diffusion. This adds spatial mixing. Default is False.
  • diffuse_on_diffuse: If True, the hyper-diffusion is computed as the diffusion of the diffusion. This adds spatial mixing. Default is False.
  • 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.

Notes:

-

Good Values:

  • There is an anlytical solution to the (inviscid, ν = 0) KdV of u(t, x) = - 1/2 c^2 sech^2(c/2 (x - ct - a)) with the hyperbolic secant sech and arbitrarily selected speed c and shift a.
  • For a nice simulation with an initial condition that breaks into solitons choose domain_extent=20.0 and an initial condition with the first 5-10 modes. Set dt=0.01, num points in the range of 50-200 are sufficient.
Source code in exponax/stepper/_korteweg_de_vries.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    convection_scale: float = -6.0,
    diffusivity: float = 0.0,
    dispersivity: float = 1.0,
    hyper_diffusivity: float = 0.01,
    advect_over_diffuse: bool = False,
    diffuse_over_diffuse: bool = False,
    single_channel: bool = False,
    conservative: bool = False,
    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}`) (hyper-viscous)
    Korteweg-de Vries equation on periodic boundary conditions.

    In 1d, the Korteweg-de Vries equation is given by

    ```
        uₜ + b₁ 1/2 (u²)ₓ + a₃ uₓₓₓ = ν uₓₓ - μ uₓₓₓₓ
    ```

    with `b₁` the convection coefficient, `a₃` the dispersion coefficient
    and `ν` the diffusivity. Oftentimes `b₁ = -6` and `ν = 0`. The
    nonlinearity is similar to the Burgers equation and the number of
    channels grow with the number of spatial dimensions. In higher
    dimensions, the equation reads (using vector format for the channels)

    ```
        uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) + a₃ 1 ⋅ (∇⊙∇⊙(∇u)) = ν Δu - μ ((∇⊙∇) ⋅ (∇⊙∇)) u
    ```

    or

    ```
        uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) + a₃ ∇ ⋅ ∇(Δu) = ν Δu - μ Δ(Δu)
    ```

    if `advect_over_diffuse` is `True` and `diffuse_on_diffuse` is `True`,

    In 1d, the expected temporal behavior is that the initial condition
    breaks into soliton waves that propagate at a speed depending on their
    height. They interact with other soliton waves by being spatially
    displaced but having an unchanged shape and propagation speed. If either
    the `diffusivity` or the `hyper_diffusivity` is non-zero (by default,
    the latter is active), the solution will decay over time with the
    produced small-scall features decaying the fastest.

    **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.
    - `convection_scale`: The convection coefficient `b₁`. Note that the
        convection is already scaled by 1/2 to account for the conservative
        evaluation. The value of `b₁` scales it further. Oftentimes `b₁ =
        -6` to match the analytical soliton solutions. See also
        https://en.wikipedia.org/wiki/Korteweg%E2%80%93De_Vries_equation#One-soliton_solution
    - `diffusivity`: The rate at which the solution decays.
    - `dispersivity`: The dispersion coefficient `a₃`. Dispersion refers
        to wavenumber-dependent advection, i.e., higher wavenumbers are
        advected faster. Default `1.0`,
    - `hyper_diffusivity`: The hyper-diffusivity (associated with a
        fourth-order term). This stepper only supports scalar (=isotropic)
        hyper-diffusivity. Default: 0.01.
    - `advect_over_diffuse`: If `True`, the dispersion is computed as
        advection over diffusion. This adds spatial mixing. Default is
        `False`.
    - `diffuse_on_diffuse`: If `True`, the hyper-diffusion is computed as
        the diffusion of the diffusion. This adds spatial mixing. Default is
        `False`.
    - `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.

    **Notes:**

        -

    **Good Values:**

    - There is an anlytical solution to the (inviscid, `ν = 0`) KdV of
        `u(t, x) = - 1/2 c^2 sech^2(c/2 (x - ct - a))` with the hyperbolic
        secant `sech` and arbitrarily selected speed `c` and shift `a`.
    - For a nice simulation with an initial condition that breaks into
        solitons choose `domain_extent=20.0` and an initial condition with
        the first 5-10 modes. Set dt=0.01, num points in the range of 50-200
        are sufficient.
    """
    self.convection_scale = convection_scale
    self.diffusivity = diffusivity
    self.dispersivity = dispersivity
    self.hyper_diffusivity = hyper_diffusivity
    self.advect_over_diffuse = advect_over_diffuse
    self.diffuse_over_diffuse = diffuse_over_diffuse
    self.single_channel = single_channel
    self.conservative = conservative
    self.dealiasing_fraction = dealiasing_fraction

    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)