Stability Analysis of Finite Difference Schemes for the Advection-Diffusion Equation

摘要

We present a collection of stability results for finite difference approximations to the advection-diffusion equation sub ut = a sub ux + b sub uxx. The results are for centered difference schemes in space and include explicit schemes in time up to fourth order and schemes that use different space and time discretizations for the advective and diffusive terms. The results are derived from a uniform framework based on the Schur-Cohn theory of simple von Neumann Polynomials and are necessary and sufficient for the stability of the Cauchy problem. Some of the results are believed to be new. (Author)