Non-normality Eeects in a Discretised Nonlinear Reaction-convection-diusion Equation

What is the long-time eeect of adding convection to a discretised reaction-diiusion equation? For linear problems, it is well known that convection may denormalise the process, and, in particular, eigenvalue-based stability predictions may be over-optimistic. This work deals with a related issue| with a nonlinear reaction term, the non-normality can greatly innuence the long-time dynamics. For a nonlinear model problem with Dirichlet boundary conditions, it is shown that the basin of attraction of the \correct" steady state can be shrunk in a directionally biased manner. A normwise analysis provides lower bounds on the basin of attraction and a more revealing picture is provided by pseudo-eigenvalues. In extreme cases, the computed solution can converge to a spurious, bounded, steady-state that exists only in nite precision arithmetic. The impact of convection on the existence and stability of spurious, periodic solutions is also quantiied.