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General Vorticity Convection Stepper¤

exponax.stepper.generic.GeneralVorticityConvectionStepper ¤

Bases: BaseStepper

Source code in exponax/stepper/generic/_vorticity_convection.py 
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class GeneralVorticityConvectionStepper(BaseStepper):
    vorticity_convection_scale: float
    linear_coefficients: tuple[float, ...]
    injection_mode: int
    injection_scale: float
    dealiasing_fraction: float

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        vorticity_convection_scale: float = 1.0,
        linear_coefficients: tuple[float, ...] = (0.0, 0.0, 0.001),
        injection_mode: int = 4,
        injection_scale: float = 0.0,
        order: int = 2,
        dealiasing_fraction: float = 2 / 3,
        num_circle_points: int = 16,
        circle_radius: float = 1.0,
    ):
        """
        Timestepper for 2D PDEs consisting of vorticity convection term and an
        arbitrary combination of (isotropic) linear derivatives.

        ```
            uₜ + b ([1, -1]ᵀ ⊙ ∇(Δ⁻¹u)) ⋅ ∇u = sum_j a_j (1⋅∇ʲ)u
        ```

        where `b` is the vorticity convection scale, `a_j` are the coefficients
        of the linear derivatives, and `∇ʲ` is the `j`-th derivative operator.

        In the default configuration, this corresponds to the 2D Navier-Stokes
        simulation with a viscosity of `ν = 0.001` (the resulting Reynols number
        depends on the `domain_extent`. In the case of a unit square domain,
        i.e., `domain_extent = 1`, the Reynols number is `Re = 1/ν = 1000`).
        Depending on the initial state, this corresponds to a decaying 2D
        turbulence.

        Additionally, one can set an `injection_mode` and `injection_scale` to
        inject energy into the system. For example, this allows for the
        simulation of forced turbulence (=Kolmogorov flow).

        **Arguments:**

        - `num_spatial_dims`: 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`: 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, 0.001)` corresponds to pure regular diffusion.
        - `vorticity_convection_scale`: The scale `b` of the vorticity
            convection term.
        - `injection_mode`: The mode of the injection.
        - `injection_scale`: The scale of the injection. Defaults to `0.0` which
            means no injection. Hence, the flow will decay over time.
        - `dealiasing_fraction`: The fraction of the modes that are kept after
            dealiasing. The default value `2/3` corresponds to the 2/3 rule.
        - `order`: The order of the ETDRK method to use. Must be one of {0, 1,
            2, 3, 4}. The option `0` only solves the linear part of the
            equation. Hence, only use this for linear PDEs. For nonlinear PDEs,
            a higher order method tends to be more stable and accurate. `2` is
            often a good compromis in single-precision. Use `4` together with
            double precision (`jax.config.update("jax_enable_x64", True)`) for
            highest accuracy.
        - `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.
        - `circle_radius`: The radius of the contour used to compute the
            coefficients of the exponential time differencing Runge Kutta
            method.
        """
        if num_spatial_dims != 2:
            raise ValueError(f"Expected num_spatial_dims = 2, got {num_spatial_dims}.")
        self.vorticity_convection_scale = vorticity_convection_scale
        self.linear_coefficients = linear_coefficients
        self.injection_mode = injection_mode
        self.injection_scale = injection_scale
        self.dealiasing_fraction = dealiasing_fraction
        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"]:
        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"],
    ) -> VorticityConvection2d:
        if self.injection_scale == 0.0:
            return VorticityConvection2d(
                self.num_spatial_dims,
                self.num_points,
                convection_scale=self.vorticity_convection_scale,
                derivative_operator=derivative_operator,
                dealiasing_fraction=self.dealiasing_fraction,
            )
        else:
            return VorticityConvection2dKolmogorov(
                self.num_spatial_dims,
                self.num_points,
                convection_scale=self.vorticity_convection_scale,
                derivative_operator=derivative_operator,
                dealiasing_fraction=self.dealiasing_fraction,
                injection_mode=self.injection_mode,
                injection_scale=self.injection_scale,
            )
__init__ ¤
__init__(
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    vorticity_convection_scale: float = 1.0,
    linear_coefficients: tuple[float, ...] = (
        0.0,
        0.0,
        0.001,
    ),
    injection_mode: int = 4,
    injection_scale: float = 0.0,
    order: int = 2,
    dealiasing_fraction: float = 2 / 3,
    num_circle_points: int = 16,
    circle_radius: float = 1.0
)

Timestepper for 2D PDEs consisting of vorticity convection term and an arbitrary combination of (isotropic) linear derivatives.

    uₜ + b ([1, -1]ᵀ ⊙ ∇(Δ⁻¹u)) ⋅ ∇u = sum_j a_j (1⋅∇ʲ)u

where b is the vorticity convection scale, a_j are the coefficients of the linear derivatives, and ∇ʲ is the j-th derivative operator.

In the default configuration, this corresponds to the 2D Navier-Stokes simulation with a viscosity of ν = 0.001 (the resulting Reynols number depends on the domain_extent. In the case of a unit square domain, i.e., domain_extent = 1, the Reynols number is Re = 1/ν = 1000). Depending on the initial state, this corresponds to a decaying 2D turbulence.

Additionally, one can set an injection_mode and injection_scale to inject energy into the system. For example, this allows for the simulation of forced turbulence (=Kolmogorov flow).

Arguments:

  • num_spatial_dims: 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: 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, 0.001) corresponds to pure regular diffusion.
  • vorticity_convection_scale: The scale b of the vorticity convection term.
  • injection_mode: The mode of the injection.
  • injection_scale: The scale of the injection. Defaults to 0.0 which means no injection. Hence, the flow will decay over time.
  • dealiasing_fraction: The fraction of the modes that are kept after dealiasing. The default value 2/3 corresponds to the 2/3 rule.
  • order: The order of the ETDRK method to use. Must be one of {0, 1, 2, 3, 4}. The option 0 only solves the linear part of the equation. Hence, only use this for linear PDEs. For nonlinear PDEs, a higher order method tends to be more stable and accurate. 2 is often a good compromis in single-precision. Use 4 together with double precision (jax.config.update("jax_enable_x64", True)) for highest accuracy.
  • 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.
  • circle_radius: The radius of the contour used to compute the coefficients of the exponential time differencing Runge Kutta method.
Source code in exponax/stepper/generic/_vorticity_convection.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    vorticity_convection_scale: float = 1.0,
    linear_coefficients: tuple[float, ...] = (0.0, 0.0, 0.001),
    injection_mode: int = 4,
    injection_scale: float = 0.0,
    order: int = 2,
    dealiasing_fraction: float = 2 / 3,
    num_circle_points: int = 16,
    circle_radius: float = 1.0,
):
    """
    Timestepper for 2D PDEs consisting of vorticity convection term and an
    arbitrary combination of (isotropic) linear derivatives.

    ```
        uₜ + b ([1, -1]ᵀ ⊙ ∇(Δ⁻¹u)) ⋅ ∇u = sum_j a_j (1⋅∇ʲ)u
    ```

    where `b` is the vorticity convection scale, `a_j` are the coefficients
    of the linear derivatives, and `∇ʲ` is the `j`-th derivative operator.

    In the default configuration, this corresponds to the 2D Navier-Stokes
    simulation with a viscosity of `ν = 0.001` (the resulting Reynols number
    depends on the `domain_extent`. In the case of a unit square domain,
    i.e., `domain_extent = 1`, the Reynols number is `Re = 1/ν = 1000`).
    Depending on the initial state, this corresponds to a decaying 2D
    turbulence.

    Additionally, one can set an `injection_mode` and `injection_scale` to
    inject energy into the system. For example, this allows for the
    simulation of forced turbulence (=Kolmogorov flow).

    **Arguments:**

    - `num_spatial_dims`: 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`: 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, 0.001)` corresponds to pure regular diffusion.
    - `vorticity_convection_scale`: The scale `b` of the vorticity
        convection term.
    - `injection_mode`: The mode of the injection.
    - `injection_scale`: The scale of the injection. Defaults to `0.0` which
        means no injection. Hence, the flow will decay over time.
    - `dealiasing_fraction`: The fraction of the modes that are kept after
        dealiasing. The default value `2/3` corresponds to the 2/3 rule.
    - `order`: The order of the ETDRK method to use. Must be one of {0, 1,
        2, 3, 4}. The option `0` only solves the linear part of the
        equation. Hence, only use this for linear PDEs. For nonlinear PDEs,
        a higher order method tends to be more stable and accurate. `2` is
        often a good compromis in single-precision. Use `4` together with
        double precision (`jax.config.update("jax_enable_x64", True)`) for
        highest accuracy.
    - `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.
    - `circle_radius`: The radius of the contour used to compute the
        coefficients of the exponential time differencing Runge Kutta
        method.
    """
    if num_spatial_dims != 2:
        raise ValueError(f"Expected num_spatial_dims = 2, got {num_spatial_dims}.")
    self.vorticity_convection_scale = vorticity_convection_scale
    self.linear_coefficients = linear_coefficients
    self.injection_mode = injection_mode
    self.injection_scale = injection_scale
    self.dealiasing_fraction = dealiasing_fraction
    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)