Discretization Schemes

In xpektra, we define a discretization scheme which allows us to correctly define various differentiation operators. To do so, we need two information:

• The underlying grid in the physical space i.e if regular, staggered, etc.
• The differentiation formula to be used.

In order to facilitate this, we define a base class Scheme which provides the necessary infrastructure to define a discretization scheme.

xpektra.scheme.Scheme

Abstract base class for a complete discretization strategy.

A Scheme is a self-contained object responsible for generating the discrete gradient operator based on a given spectral space.

compute_gradient_operator(wavenumbers_mesh: list[jax.jaxlib._jax.Array]) -> jaxlib._jax.Array

The primary output of any scheme. The gradient operator field has shape ( (N,)dim, (dim,)rank).

is_compatible(transform) -> bool

Checks if the scheme is compatible with the given transform.

The Scheme class is a base class for all the discretization schemes. One can create different discretization schemes by subclassing the Scheme class and implementing the formula method. The formula method should return the gradient operator field for a given wavenumber and grid spacing. In xpektra, we have implemented the CartesianScheme which takes a regular grid in physical space and returns the gradient operator field in spectral space.

xpektra.scheme.DiagonalScheme

Base class for schemes operating on a uniform Cartesian grid where the differentiation is diagonal in Fourier space.

__init__(space: SpectralSpace)

apply_gradient(u_hat: jaxlib._jax.Array) -> jaxlib._jax.Array

Applies the gradient operator on the fly. Computes: grad_hat_ij = Dξ_i * u_hat_j

apply_symmetric_gradient(u_hat: jaxlib._jax.Array) -> jaxlib._jax.Array

Applies the symmetric gradient operator on the fly. Computes: eps_hat_ij = 0.5 * (Dξ_i * u_hat_j + Dξ_j * u_hat_i)

apply_divergence(u_hat: jaxlib._jax.Array) -> jaxlib._jax.Array

Applies the divergence operator on the fly. Computes: div_hat_i = Dξ_j * u_hat_ji

apply_laplacian(u_hat: jaxlib._jax.Array) -> jaxlib._jax.Array

Applies the Laplacian operator on the fly. Computes: lap_hat = -|Dξ|^2 * u_hat

compute_gradient_operator(wavenumbers_mesh) -> jaxlib._jax.Array

Builds the full gradient operator field using the scheme's formula.

is_compatible(transform)

formula(xi, dx, iota, factor)

The core formula for the discrete derivative in Fourier space. Must be implemented by concrete schemes.

To define the differentiation formula, we need to implement the formula method. The formula method should return the gradient operator field for a given wavenumber and grid spacing. In xpektra, we have various differentiation schemes available which can be used to define the differentiation formula.

xpektra.scheme.FourierScheme

Class implementing the standard spectral 'Fourier' derivative.

formula(xi, dx, iota, factor)

The formula is given by:

\[D(\xi) = \iota \xi\]

where \(\iota\) is the imaginary unit and \(\xi\) is the wavenumber.

xpektra.scheme.CentralDifference

Implements the standard central difference scheme.

formula(xi, dx, iota, factor)

The formula is given by:

\[D(\xi) = \iota \frac{\sin(\xi \Delta x)}{\Delta x}\]

where \(\iota\) is the imaginary unit, \(\xi\) is the wavenumber and \(\Delta x\) is the grid spacing.

xpektra.scheme.ForwardDifference

Implements the forward difference scheme.

formula(xi, dx, iota, factor)

The formula is given by:

\[D(\xi) = \frac{\exp(\iota \xi \Delta x) - 1}{\Delta x}\]

where \(\iota\) is the imaginary unit, \(\xi\) is the wavenumber and \(\Delta x\) is the grid spacing.

xpektra.scheme.BackwardDifference

Implements the backward difference scheme.

formula(xi, dx, iota, factor)

The formula is given by:

\[D(\xi) = \frac{1 - \exp(-\iota \xi \Delta x)}{\Delta x}\]

where \(\iota\) is the imaginary unit, \(\xi\) is the wavenumber and \(\Delta x\) is the grid spacing.

xpektra.scheme.RotatedDifference

Implements the rotated finite difference scheme (Willot/HEX8R).

formula(xi, dx, iota, factor)

The formula is given by:

\[D(\xi) = \frac{2 \iota \tan(\xi \Delta x / 2) \Delta x}{2 \Delta x}\]

where \(\iota\) is the imaginary unit, \(\xi\) is the wavenumber and \(\Delta x\) is the grid spacing.

xpektra.scheme.FourthOrderCentralDifference

Implements the fourth order difference scheme.

formula(xi, dx, iota, factor)

The formula is given by:

\[D(\xi) = \iota \frac{8 \sin(\xi \Delta x) - 2 \sin(2 \xi \Delta x) + 8 \sin(3 \xi \Delta x) - \sin(4 \xi \Delta x)}{6 \Delta x}\]

where \(\iota\) is the imaginary unit, \(\xi\) is the wavenumber and \(\Delta x\) is the grid spacing.

xpektra.scheme.SixthOrderCentralDifference

Implements the sixth order difference scheme.

formula(xi, dx, iota, factor)

The formula is given by:

\[D(\xi) = \iota \frac{9 \sin(\xi \Delta x) - 3 \sin(2 \xi \Delta x) + \sin(3 \xi \Delta x)}{6 \Delta x}\]

where \(\iota\) is the imaginary unit, \(\xi\) is the wavenumber and \(\Delta x\) is the grid spacing.

xpektra.scheme.EighthOrderCentralDifference

Implements the eighth order difference scheme.

formula(xi, dx, iota, factor)

The formula is given by:

\[D(\xi) = \iota \frac{8 \sin(\xi \Delta x) - 2 \sin(2 \xi \Delta x) + 8 \sin(3 \xi \Delta x) - \sin(4 \xi \Delta x)}{12 \Delta x}\]

where \(\iota\) is the imaginary unit, \(\xi\) is the wavenumber and \(\Delta x\) is the grid spacing.