Partially implicit runge-kutta methods for wave-like equations

Abstract

Runge-Kutta methods are used to integrate in time systems of differential equations. Implicit methods are designed to overcome numerical instabilities appearing during the evolution of a system of equations. We will present partially implicit Runge-Kutta methods for a particular structure of equations, generalization of a wave equation; the partially implicit term refers to this structure, where the implicit term appears only in a subset of the system of equations. These methods do not require any inversion of operators and the computational costs are similar to those of explicit Runge-Kutta methods. Partially implicit Runge-Kutta methods are derived up to third-order of convergence. We analyze their stability properties and show the practical applicability in several numerical examples.