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

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 combination of stability, minimal filtering, simplicity, and freedom from spurious noise for integrating the nonhydrostatic or hydrostatic equations. Schemes wherein the fast modes are integrated implicitly and the slow modes explicitly are also analyzed. These semi-implicit schemes can be used with a greater variety of advection schemes than the explicit time-split approaches and generally require less filtering than the split-explicit schemes for stability. However, a multidimensional elliptic equation must be solved with each time step. For nonhydrostatic elastic models using the KW time-split method, an acoustic filter is presented that allows a reduction of previously necessary filtering in the KW scheme, and a method for integrating the buoyancy equation is discussed that results in the large time step being limited by a Courant condition based on the advection velocity and not on the fastest gravity-wave speed.