Spatially dispersionless, unconditionally stable FC-AD solvers for variable-coefficient PDEs

TL;DR

This paper introduces fast, stable, high-order solvers for variable-coefficient PDEs that eliminate spatial dispersion and leverage Fourier continuation and an alternating direction strategy for efficient computation.

Contribution

The authors develop a novel combination of Fourier continuation and preconditioned solvers with an alternating direction approach for high-order, unconditionally stable PDE solutions in complex domains.

Findings

Achieve spatial dispersionless solutions with high-order accuracy.

Demonstrate efficiency with FFT-based computation.

Successfully apply to diffusion and wave problems in heterogeneous media.

Abstract

We present fast, spatially dispersionless and unconditionally stable high-order solvers for Partial Differential Equations (PDEs) with variable coefficients in general smooth domains. Our solvers, which are based on (i) A certain "Fourier continuation" (FC) method for the resolution of the Gibbs phenomenon, together with (ii) A new, preconditioned, FC-based solver for two-point boundary value problems (BVP) for variable-coefficient Ordinary Differential Equations, and (iii) An Alternating Direction strategy, generalize significantly a class of FC-based solvers introduced recently for constant-coefficient PDEs. The present algorithms, which are applicable, with high-order accuracy, to variable-coefficient elliptic, parabolic and hyperbolic PDEs in general domains with smooth boundaries, are unconditionally stable, do not suffer from spatial numerical dispersion, and they run at FFT speeds. The accuracy, efficiency and overall capabilities of our methods are demonstrated by means of applications to challenging problems of diffusion and wave propagation in heterogeneous media.