Linear Instability Implies Spurious Periodic Solutions

Abstract

We analyse discrete approximations of reaction-diffusion-convection equations and show that linearized instability implies the existence of spurious periodic solutions in the fully nonlinear problem. The result is proved by using ideas from bifurcation theory. Using singularity theory we provide a precise local description of the spurious solutions. The results form the basis for an analysis of the range of discretization parameters in which spurious solutions can exist, their magnitude, and their spatial structure. We present a modified equations approach to determine criteria under which spurious periodic solutions exist for arbitrarily small values of the time-step. The theoretical results are applied to a specific example.