On the stability of a class of splitting methods for integro-differential equations

Authors:
A. Araújo;J. R. Branco;J. A. Ferreira
Affiliations:
CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, 3000 Coimbra, Portugal;Department of Physics and Mathematics, Insituto Superior de Engenharia de Coimbra, Rua Pedro Nunes-Quinta da Nora, 3030-199 Coimbra, Portugal;CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, 3000 Coimbra, Portugal
Venue:
Applied Numerical Mathematics
Year:
2009

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Abstract

The classical convection-diffusion-reaction equation has the unphysical property that if a sudden change in the dependent variable is made at any point, it will be felt instantly everywhere. This phenomena violate the principle of causality. Over the years, several authors have proposed modifications in an effort to overcome the propagation speed defect. The purpose of this paper is to study, from analytical and numerical point of view a modification to the classical model that take into account the memory effects. Besides the finite speed of propagation, we establish an energy estimate to the exact solution. We also present a numerical method which has the same qualitative property of the exact solution. Finally we illustrate the theoretical results with some numerical simulations.