High-order low-storage explicit Runge-Kutta schemes for equations with quadratic nonlinearities

Abstract

We show in this paper that third- and fourth-order low storage Runge-Kutta algorithms can be built specifically for quadratic nonlinear operators, at the expense of roughly doubling the time needed for evaluating the temporal derivatives. The resulting algorithms are especially well suited for computational fluid dynamics. Examples are given for the H\'enon-Heiles Hamiltonian system and, in one and two space dimensions, for the Burgers equation using both a pseudo-spectral code and a spectral element code, respectively. The scheme is also shown to be practical in three space solving the incompressible Euler equation using a fully parallelized pseudo-spectral code.