The Stability of Time-Split Numerical Methods for the Hydrostatic and the Nonhydrostatic Elastic Equations

Abstract

Abstract The mathematical equivalence of the linearized two-dimensional (2D) shallow-water system and the 2D acoustic-advection system strongly suggests that time-split schemes designed for the hydrostatic equations can be employed in nonhydrostatic models and vice versa. Stability analyses are presented for several time-split numerical methods for integrating the two systems. The primary interest is in the nonhydrostatic system and in explicit numerical schemes where no multidimensional elliptic equations arise; thus, a detailed analysis of the Klemp and Wilhelmson (KW) explicit technique for integrating the time-split nonhydrostatic system is undertaken. It is found that the interaction between propagating and advecting acoustic modes can introduce severe constraints on the maximum allowable time steps. Proper filtering can remove these constraints. Other explicit time-split schemes are analysed, and, of all the explicit schemes considered, it is believed that the KW time-split method offers the best co...