Splitting methods for mixed hyperbolic-parabolic systems / Alf Gerisch, David F. Griffiths, Rüdiger Weiner, and Mark A. J. Chaplain

Abstract

In this paper we present two different approaches to the numerical solution of a system of coupled parabolic{hyperbolic partial differential equations (PDEs). The first approach uses a finite difference scheme based on a splitting of the PDE system. This method is computationally inexpensive but is prone to generating spurious oscillations in the solution in certain regions of the spatial domain. These lead to incorrect negative solution values. The second approach uses the method of lines with special attention being given to preserving the positivity of the solution. Here we use splitting techniques for the solution of the resulting system of stiff ordinary differential equations. The performance of the two approaches on a model problem from mathematical biology is compared and discussed and conclusions drawn.