Explicit time stepping methods with high stage order and monotonicity properties

Abstract

This paper introduces a three and a four order explicit time stepping method. These methods have high stage order and favorable monotonicity properties. The proposed methods are based on multistage-multistep (MM) schemes that belong to the broader class of general linear methods, which are generalizations of both Runge-Kutta and linear multistep methods. Methods with high stage order alleviate the order reduction occurring in explicit multistage methods due to non-homogeneous boundary/source terms. Furthermore, the MM schemes presented in this paper can be expressed as convex combinations of Euler steps. Consequently, they have the same monotonicity properties as the forward Euler method. This property makes these schemes well suited for problems with discontinuous solutions. © 2009 Springer Berlin Heidelberg.