A new perspective on spectral analysis of numerical schemes

摘要

Spectral analysis is an essential tool for analysing the stability and accuracy of numerical schemes for solving partial differential equations on regular meshes. In particular, spectral analysis allows a detailed study of the dispersion error, as well as anisotropic effects introduced by the mesh. When performing this analysis, many authors assume that the waves making up the solution are always orientated in the same direction as the partial differential equation's characteristics. While this is a valid assumption in some cases, it is not correct in other situations, especially for analysis of the convection-diffusion equation and similar transport phenomena. This paper addresses this issue, and resolves some long-standing misconceptions resulting from it. In particular, it is shown that for convection simulations on a regular mesh of squares, the overall level of dispersion error is not affected by the convection direction.