The effect of freezing and discretization to the asymptotic stability of relative equilibria
The purpose of the paper is to analyze numerical methods for the approximation of relative equilibria of parabolic systems in one space dimension: \[ u_t=Au_{xx}+f(u,u_x) \] In the first section the author gives a short introduction to the method of freezing relative equilibria and state conditions which ensure the asymptotic phase of these solutions. The second section concerns the numerical approximation. Thus, he introduces the finite difference approximation and states the main stability result for the solution of the discretized equations, given by the fact that nonlinear stability properties are inherited by the numerical approximation with finite differences on a finite equidistant grid with appropriate boundary conditions. The third section focuses on the stability of the nonlinear discretized system and proves the main stability result obtained in the previous section, using resolvent estimates. The fourth section proves resolvent estimates for the discretized system. The fifth section contains some numerical examples concerning the cubic quintic Ginzburg Landau equation, which shows a variety of coherent structures: stable pulse, fronts, sources, sinks. By a counterexample for the Nagumo equation the author shows that some of the assumptions on the boundary operators are sharp.
partial differential algebraic equations
scientific article; zbMATH DE number 5281921