Método do fator de integração implícito para problemas de reação-difusão

Abstract

Reaction-Diffusion problems are mathematical models that describe phenomena observed in various applications of Physics, Chemistry, Materials Science and Biology. In such cases, we can use the method of implicit integration factor (IIF), which decouples the terms of diffusion and reaction in order to calculate explicity the diffusive terms and treat implicitly reactive terms. The computational cost of the IIF (storage and processing) makes this method not very attractive and one of the approaches to work around this problem is to employ strategies approaches using the Krylov subspace approximations to reduce arithmetic operations for the evaluation of the exponential matrix involved in this process. Another approach is to work with the compact representation of the spatial discretization to obtain the compact implicit integration factor method, with reduced costs of storage and processing then those of IIF method. In this paper, we present this procedure along with computational experiments in two and three dimensional domains for different equations in order to test the effectiveness of each method. Application examples of the procedure are linear reaction-diffusion problems, Allen-Cahn, Ginzburg Landau Schnackenberg FitzHugh-Nagumo and discussed in order to demonstrate the applicability of the method.