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Burgers¤

In 1D:

\[ \frac{\partial u}{\partial t} + \frac{1}{2} \frac{\partial u^2}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2} \]

In higher dimensions:

\[ \frac{\partial u}{\partial t} + \frac{1}{2} \nabla \cdot (u \odot u) = \nu \nabla \cdot \nabla u \]

(with as many channels (=velocity components) as spatial dimensions)

exponax.stepper.Burgers ¤

Bases: BaseStepper

Source code in exponax/stepper/_burgers.py 
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class Burgers(BaseStepper):
    diffusivity: 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,
        *,
        diffusivity: float = 0.1,
        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}`) Burgers equation on
        periodic boundary conditions.

        In 1d, the Burgers equation is given by

        ```
            uₜ + b₁ 1/2 (u²)ₓ = ν uₓₓ
        ```

        with `b₁` the convection coefficient and `ν` the diffusivity. Oftentimes
        `b₁ = 1`. 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 channels grow with the dimension, i.e. in 2d
        the state `u` is represented by a tensor of shape `(2, num_points,
        num_points)`. The equation in 2d reads (using vector format for the two
        channels)

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

        with `∇ ⋅` the divergence operator and `Δ` the Laplacian.

        The expected temporal behavior is that the initial condition becomes
        "sharper"; in 1d positive values move to the right and negative values
        to the left. Smooth shocks develop that propagate at speed depending on
        the height difference. Ultimately, the solution decays to a constant
        state.

        **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` in the Burgers equation.
            Default: 0.1.
        - `convection_scale`: The scaling factor for the convection term.
            Note that the scaling by 1/2 is always performed. Default: 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.

        **Notes:**

        - If the `diffusivity` is set too low, spurious oscillations may
            occur because the solution becomes "too discontinous". Such
            simulations are not possible with Fourier pseudospectral methods.
            Sometimes increasing the number of points `N` can help.

        **Good Values:**

        - Next to the defaults of `diffusivity=0.1` and
            `convection_scale=1.0`, the following values are good starting
            points:
            - `num_points=100` for 1d.
            - `domain_extent=1`
            - `dt=0.1`
            - A bandlimited initial condition with maximum absolute value of
                ~1.0
        """
        self.diffusivity = diffusivity
        self.convection_scale = convection_scale
        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"]:
        # The linear operator is the same for all D channels
        return self.diffusivity * build_laplace_operator(derivative_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,
    *,
    diffusivity: float = 0.1,
    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}) Burgers equation on periodic boundary conditions.

In 1d, the Burgers equation is given by

    uₜ + b₁ 1/2 (u²)ₓ = ν uₓₓ

with b₁ the convection coefficient and ν the diffusivity. Oftentimes b₁ = 1. 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 channels grow with the dimension, i.e. in 2d the state u is represented by a tensor of shape (2, num_points, num_points). The equation in 2d reads (using vector format for the two channels)

    uₜ + b₁ 1/2 ∇ ⋅ (u ⊗ u) = ν Δu

with ∇ ⋅ the divergence operator and Δ the Laplacian.

The expected temporal behavior is that the initial condition becomes "sharper"; in 1d positive values move to the right and negative values to the left. Smooth shocks develop that propagate at speed depending on the height difference. Ultimately, the solution decays to a constant state.

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 in the Burgers equation. Default: 0.1.
  • convection_scale: The scaling factor for the convection term. Note that the scaling by 1/2 is always performed. Default: 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.

Notes:

  • If the diffusivity is set too low, spurious oscillations may occur because the solution becomes "too discontinous". Such simulations are not possible with Fourier pseudospectral methods. Sometimes increasing the number of points N can help.

Good Values:

  • Next to the defaults of diffusivity=0.1 and convection_scale=1.0, the following values are good starting points:
    • num_points=100 for 1d.
    • domain_extent=1
    • dt=0.1
    • A bandlimited initial condition with maximum absolute value of ~1.0
Source code in exponax/stepper/_burgers.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    diffusivity: float = 0.1,
    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}`) Burgers equation on
    periodic boundary conditions.

    In 1d, the Burgers equation is given by

    ```
        uₜ + b₁ 1/2 (u²)ₓ = ν uₓₓ
    ```

    with `b₁` the convection coefficient and `ν` the diffusivity. Oftentimes
    `b₁ = 1`. 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 channels grow with the dimension, i.e. in 2d
    the state `u` is represented by a tensor of shape `(2, num_points,
    num_points)`. The equation in 2d reads (using vector format for the two
    channels)

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

    with `∇ ⋅` the divergence operator and `Δ` the Laplacian.

    The expected temporal behavior is that the initial condition becomes
    "sharper"; in 1d positive values move to the right and negative values
    to the left. Smooth shocks develop that propagate at speed depending on
    the height difference. Ultimately, the solution decays to a constant
    state.

    **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` in the Burgers equation.
        Default: 0.1.
    - `convection_scale`: The scaling factor for the convection term.
        Note that the scaling by 1/2 is always performed. Default: 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.

    **Notes:**

    - If the `diffusivity` is set too low, spurious oscillations may
        occur because the solution becomes "too discontinous". Such
        simulations are not possible with Fourier pseudospectral methods.
        Sometimes increasing the number of points `N` can help.

    **Good Values:**

    - Next to the defaults of `diffusivity=0.1` and
        `convection_scale=1.0`, the following values are good starting
        points:
        - `num_points=100` for 1d.
        - `domain_extent=1`
        - `dt=0.1`
        - A bandlimited initial condition with maximum absolute value of
            ~1.0
    """
    self.diffusivity = diffusivity
    self.convection_scale = convection_scale
    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)