Abstract
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.
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