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Convergence of the Peaceman-Rachford approximation for reaction-diffusion systems. (English) Zbl 1034.65077
Reaction-diffusion systems appear in many different situations, in chemistry, in biology. A mathematical model for pattern formation is presented in this article. These systems are of the form $u_t - M \Delta u +F(u) = 0,$ where $$M$$ is a $$m \times m$$ matrix whose spectrum is included in $$\{ \operatorname{Re} z > 0\}$$. Time discretization of this problem is considered, especially the Peaceman-Rachford formula is investigated. This approximation is defined by: $P(t)= ({\mathbf 1} +t F/2)^{-1}({\mathbf 1} +t M\Delta /2) ({\mathbf 1} -t M\Delta /2)^{-1} ({\mathbf 1} -t F/2) .$ First, stability of Peaceman-Rachford scheme is shown and then convergence of this scheme to the exact solution is proved. Moreover it is shown that this convergence is of order two. As an examples for numerical presentation two models are used. The first is a model of Turing system where the spatial structures are created by an interaction of nonlinear phenomena and spectral properties, and the second is the Ginzburg-Landau equation, where the solution is a complex function and coefficients belong to a complex plane, too. It is alsao explained how to approximate the operator $$({\mathbf 1} +t F/2)^{-1}$$ without solving a nonlinear system.

##### MSC:
 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35K57 Reaction-diffusion equations 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
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