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Classroom Note: Stability Considerations for Numerical Methods

In this paper we address the issue of stability in numerical methods for partial differential equations (PDEs) through normal mode analysis of a particular scheme. It is shown that when numerical methods appear to be stable, there are subtleties one must be aware of before conclusions may be drawn. This is illustrated through the analysis of the one-way wave equation using the leap-frog scheme. A bifurcation from stability to instability is illustrated along with numerical simulations illustrating the growing modes.