On balanced approximations for time integration of multiple time scale systems

The effect of various numerical approximations used to solve linear and nonlinear problems with multiple time scales is studied in the framework of modified equation analysis (MEA). First, MEA is used to study the effect of linearization and splitting in a simple nonlinear ordinary differential equation (ODE), and in a linear partial differential equation (PDE). Several time discretizations of the ODE and PDE are considered, and the resulting truncation terms are compared analytically and numerically. It is demonstrated quantitatively that both linearization and splitting can result in accuracy degradation when a computational time step larger than any of the competing (fast) time scales is employed. Many of the issues uncovered on the simple problems are shown to persist in more realistic applications. Specifically, several differencing schemes using linearization and/or time splitting are applied to problems in nonequilibrium radiation–diffusion, magnetohydrodynamics, and shallow water flow, and their solutions are compared to those using balanced time integration methods.