Nonstandard Fourier Pseudospectral Time Domain (PSTD) Schemes for Partial Differential Equations

TL;DR

This paper introduces nonstandard Fourier pseudospectral time domain schemes that combine spectral methods with modified finite differences for solving various hyperbolic and parabolic PDEs, achieving unconditional stability.

Contribution

It presents a novel class of PSTD schemes using dispersion relation-based correction factors in k-space, applicable to any space dimension and PDE type.

Findings

Schemes are unconditionally stable.

Applicable to wave, diffusion, and convection-diffusion equations.

Demonstrated effectiveness in multiple PDE examples.

Abstract

A class of nonstandard pseudospectral time domain (PSTD) schemes for solving time-dependent hyperbolic and parabolic partial differential equations (PDEs) is introduced. These schemes use the Fourier collocation spectral method to compute spatial gradients and a nonstandard finite difference scheme to integrate forwards in time. The modified denominator function that makes the finite difference time scheme exact is transformed into the spatial frequency domain or k-space using the dispersion relation for the governing PDE. This allows the correction factor to be applied in the spatial frequency domain as part of the spatial gradient calculation. The derived schemes can be formulated to be unconditionally stable, and apply to PDEs in any space dimension. Examples of the resulting nonstandard PSTD schemes for several PDEs are given, including the wave equation, diffusion equation, and convection-diffusion equation.