A high-order conservative Patankar-type discretisation for stiff systems of production–destruction equations Original Research Article

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.