Stability Analysis of Operator Splitting for Large-Scale Ocean Modeling

Abstract

The ocean plays a crucial role in the earth's climate system, and an improved understanding of that role will be aided greatly by high-resolution simulations of global ocean circulation over periods of many years. For such simulations the computational requirements are extremely demanding and maximum efficiency is essential. However, the governing equations typically used for ocean modeling admit wave velocities having widely varying magnitudes, and this situation can create serious problems with the efficiency of numerical algorithms. One common approach to resolving these problems is to split the fast and slow dynamics into separate subproblems. The fast motions are nearly independent of depth, and it is natural to try to model these motions with a two-dimensional system of equations. These fast equations could be solved with an implicit time discretization or with an explicit method with short time steps. The slow motions would then be modeled with a three-dimensional system that is solved explicitly with long time steps that are determined by the slow wave speeds. However, if the splitting is inexact, then the equations that model the slow motions might actually contain some fast components, so the stability of explicit algorithms for the slow equations could come into doubt. In this paper we discuss some general features of the operator splitting problem, and we then describe an example of such a splitting and show that instability can arise in that case.