A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations

Authors:
Hans Burchard;Eric Deleersnijder;Andreas Meister
Affiliations:
Baltic Sea Research Institute Warnemünde, Seestr. 15, D-18119 Rostock-Warnemünde, Germany;Institut d' Astronomie et de Géophysique G. Lemaître, Université Catholique de Louvain, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium;Institute of Mathematics, University of Lübeck, Wallstrasse 40, D-23560 Lübeck, Germany
Venue:
Applied Numerical Mathematics
Year:
2003

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Abstract

In the present paper, numerically robust, unconditionally positive and conservative schemes for the discretisation of stiff systems of production-destruction equations are designed. Such model systems do typically arise in geobiochemical modelling where the reproduction of these properties is vital. We suggest modified Patankar-type methods of first- and second-order in time and compare their performance by means of approximating simple linear and non-linear model problems. For the non-linear model problem, a hybrid method combining the classical Runge-Kutta scheme with a modified Patankar-type scheme gives the best numerical approximation. The classical Robertson test problem for chemical reactions which is known for its stiffness is excellently approximated with the modified Patankar-type scheme. The procedure with respect to the derivation and analysis of the modified Patankar-type schemes can be used as a guideline to develop even unconditionally positive, conservative and third-order as well as higher order methods.