Splitting Methods for Problems with Different Timescales

Abstract

Abstract The time step for the leapfrog scheme for a symmetric hyperbolic system with multiple timescales is limited by the Courant-Friedlichs-Lewy condition based on the fastest speed present. However, in many physical cases, most of the energy is in the slowest wave, and for this wave the use of the above time step implies that the time truncation error is much smaller than the spatial truncation error. A number of methods have been proposed to overcome this imbalance—for example, the semi-implicit method and the additive splitting technique originally proposed by Marchuk with variations attributable to Strang, and Klemp and Wilhelmson. An analysis of the Marchuk splitting method for multiple timescale systems shows that if a time step based on the slow speed is used, the accuracy of the method cannot be proved, and in practice the method is quite inaccurate. If a time step is chosen that is between the two extremes, then the Klemp and Wilhelmson method can be used, but only if an ad hoc stabilization m...