Nonlinear Instability in Dissipative Finite Difference Schemes

A unified analysis of reaction-diffusion equations and their finite difference representations is presented. The parallel treatment of the two problems shows clearly when and why the finite difference approximations break down. The approach used provides a general framework for the analysis and interpretation of numerical instability in approximations of dissipative nonlinear partial differential equations Continuous and discrete problems are studied from the perspective of bifurcation theory, and numerical instability is shown to be associated with the bifurcation of periodic orbits in discrete systems. An asymptotic approach, due to Newell (SIAM J. Appl. Math., 33 (1977), 133–160), is used to investigate the instability phenomenon further. In particular, equations are derived that describe the interaction of the dynamics of the partial differential equation with the artefacts of the discretization.