Efficient low-storage Runge-Kutta schemes with optimized stability regions

A variety of numerical calculations, especially when considering wave propagation, are based on the method-of-lines, where time-dependent partial differential equations (PDEs) are first discretized in space. For the remaining time-integration, low-storage Runge-Kutta schemes are particularly popular due to their efficiency and their reduced memory requirements. In this work, we present a numerical approach to generate new low-storage Runge-Kutta (LSRK) schemes with optimized stability regions for advection-dominated problems. Adapted to the spectral shape of a given physical problem, those methods are found to yield significant performance improvements over previously known LSRK schemes. As a concrete example, we present time-domain calculations of Maxwell's equations in fully three-dimensional systems, discretized by a discontinuous Galerkin approach. (C) 2011 Elsevier Inc. All rights reserved.