Nonnormality Effects in a Discretised Nonlinear Reaction-Convection-Diffusion Equation

to diffusion, a convection term is present. Our overall aim is to look at the effect of convection on the existence and What is the long-time effect of adding convention to a discretised reaction-diffusion equation? For linear problems, it is well known stability of the true and spurious fixed points. that convection may denormalise the process, and, in particular, The potential denormalising effect of a convection term eigenvalue-based stability predictions may be overoptimistic. This can dramatically influence the behaviour close to a true work deals with a related issue—with a nonlinear reaction term, fixed point. In particular, a superficial eigenvalue test may the nonnormality can greatly influence the long-time dynamics. For be inappropriate, leading to overoptimistic predictions. In a nonlinear model problem with Dirichlet boundary conditions, it is shown that the basin of attraction of the ‘‘correct’’ steady state Section 2 we perform a ‘‘local attractivity’’ analysis and can be shrunk in a directionally biased manner. A normwise analysis show that the nonnormality causes the corresponding provides lower bounds on the basin of attraction and a more reveallower bound on the basin of attraction to become tiny. In ing picture is provided by pseudo-eigenvalues. In extreme cases, Section 3, we introduce some alternative analysis, based the computed solution can converge to a spurious, bounded, steady state that exists only in finite precision arithmetic. The impact of on pseudo-eigenvalues, that allows qualitative predictions convection on the existence and stability of spurious, periodic soluto be made about the largest practical time step. The nontions is also quantified. Q 1996 Academic Press, Inc. normality can also lead to spurious steady states that do not exist in exact arithmetic, but which arise in the presence of rounding errors. Section 4 provides examples and analy