Discretization methods with analytical solutions for a convection-reaction equation with higher-order discretizations.

Juergen Geiser

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 25 pp.

MSC 2000

35K15 Initial value problems for second-order, parabolic equations

35K57 Reaction-diffusion equations

Abstract
We introduce an improved second-order discretization method for the convection-reaction equation by combining analytical and numerical solutions. The method is derived from Godunov's scheme, see [Godunov 1959] and [Leveque 2002], and uses analytical solutions to solve the one-dimensional convection-reaction equation. We can also generalize the second-order methods for discontinuous solutions, because of the analytical test functions. One-dimensional solutions are used in the higher-dimensional solution of the numerical method. The method is based on the flux-based characteristic methods and is an attractive alternative to the classical higher-order TVD-methods, see [Harten 1983]. In this article we will focus on the derivation of analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods. For the analytical solution, we use the Laplace transformation to reduce the equation to an ordinary differential equation. With general initial conditions, e.g. spline functions, the Laplace transformation is accomplished with the help of numerical methods. The proposed discretization method skips the classical error between the convection and reaction equation by using the operator-splitting method.At the end of the article, we illustrate the higher-order method for different benchmark problems. Finally, the method is shown to produce realistic results.