Time Explicit Schemes and Spatial Finite Differences Splittings

Abstract

In this article, we conjugate time marching schemes with Finite Differences splittings into low and high modes in order to build fully explicit methods with enhanced temporal stability for the numerical solutions of PDEs. The main idea is to apply explicit schemes with less restrictive stability conditions to the linear term of the high modes equation, in order that the allowed time step for the temporal integration is only determined by the low modes. These conjugated schemes were developed in l10r for the spectral case and here we adapt them to the Finite Differences splittings provided by Incremental Unknowns, which steems from the Inertial Manifolds theory. We illustrate their improved capabilities with numerical solutions of Burgers equations, with uniform and nonuniform meshes, in dimensions one and two, when using modified Forward–Euler and Adams–Bashforth schemes. The resulting schemes use time steps of the same order of those used by semi-implicit schemes with comparable accuracy and reduced computational costs.