Accuracy Considerations of Time-Splitting Methods for Models Using Two-Time-Level Schemes

Abstract

In their recent paper, Wicker and Skamarock (2002, hereafter WS) describe a novel adaptation of a timesplitting method (Klemp and Wilhelmson 1978) for integrating the elastic equations but applied to a basic explicit time integration scheme of Runge–Kutta type. The basis for their time integration method is an explicit three-stage, one-step (two time level) scheme that they refer to as a “third-order Runge–Kutta” scheme. The conventional second-order Runge–Kutta and Adams–Bashforth schemes are both formally unstable when applied to pure oscillatory linear modes without artificial damping (Gear 1971). The explicit but economical “leapfrog” or “midpoint” method is unstable for damped modes and, in practice, requires the intervention of a time filter (Asselin 1972) to control the otherwise undamped computational “parasitic” mode in wave simulations. The method of WS, in contrast, has excellent properties of stability for both oscillatory and damped modes and, with the splitting method they propose, permits the efficient integration of a fully elastic and nonhydrostatic atmospheric model. However, as we show below, a careful analysis of the basic method they propose upon which they build their splitting scheme reveals that it is not strictly accurate to third order (except for the very special case of purely linear dynamics, where its response does exactly match that of any true three-stage third-order Runge–Kutta method) but is better thought of as a three-stage second-order method with superior characteristics of stability and accuracy. While alternative fully third-order schemes of Runge–Kutta type are discussed, the difficulties of combining these with robust splitting methods may preclude their practical implementation in numerical weather prediction models.