A new operator splitting method for the numerical solution of partial differential equations

Abstract

We present a new method of operator splitting for the numerical solution of a class of partial differential equations, 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 two less storage capacity. In the case of Hamiltonian partial differential equations the schemes we present have the additional advantage that they are symplectic. We illustrate our scheme for three nonlinear wave equations in one dimension: the shallow water equations, the Boussinesq equation, and the KdV equation. The odd-even splitting is also applicable also to PDE's in higher spatial dimension.