Nonstandard Fourier Pseudospectral Time Domain (PSTD) Schemes for Partial Differential Equations (Communications in Computational Physics, 2018)
A class of nonstandard pseudospectral time domain (PSTD) schemes forsolving time-dependent hyperbolic and parabolic partial differential equations (PDEs)is introduced. These schemes use the Fourier collocation spectral method to computespatial gradients and a nonstandard finite difference scheme to integrate forwards intime. The modified denominator function that makes the finite difference time schemeexact is transformed into the spatial frequency domain or k-space using the dispersionrelation for the governing PDE. This allows the correction factor to be appliedin the spatial frequency domain as part of the spatial gradient calculation. The derivedschemes can be formulated to be unconditionally stable, and apply to PDEs inany space dimension. Examples of the resulting nonstandard PSTD schemes for severalPDEs are given, including the wave equation, diffusion equation, and convection-diffusionequation.