Spectral implementation of a new operator splitting method for solving partial differential equations

Abstract

We extend and review a method of operator splitting for the numerical solution of a class of partial differential equations introduced by us recently, which we call the odd–even splitting. This method extends the applicability of higher order composition methods that have appeared recently in the literature. These composition methods can be considerably more efficient than conventional methods and additionally require at least a factor of 2 less storage capacity. In addition, in some cases, it is possible to use an explicit treatment of certain terms that need to be treated implicitly in other time stepping schemes. In the case of canonical Hamiltonian partial differential equations the schemes we present have the additional advantage that they are symplectic. First we will illustrate the method with an example given previously, namely the nonlinear shallow water wave equation in one dimension. In this case spatial derivatives are approximated by finite differences. The bulk of the paper is devoted to the nontrivial examples of the Euler and Navier–Stokes equations in two dimensions. Here we demonstrate how our method is extended to higher dimensional systems and also how spectral methods for computation of spatial derivatives can be incorporated into the odd–even splitting scheme. We also present a method by which explicit treatment of diffusion terms becomes possible, while retaining the advantages of implicit methods, such as stability and the ability to take larger time steps. © 1995 American Institute of Physics.