Stefan Kopecz

Stefan Kopecz, Andreas Meister

In \cite{BDM2003} the modified Patankar-Euler and modified Patankar-Runge-Kutta schemes were introduced to solve positive and conservative systems of ordinary differential equations. These modifications of the forward Euler scheme and Heun's method guarantee positivity and conservation irrespective of the chosen time step size. In this paper we introduce a general definition of modified Patankar-Runge-Kutta schemes and derive necessary and sufficient conditions to obtain first and second order methods. We also introduce two novel families of second order modified Patankar-Runge-Kutta schemes.

Stefan Kopecz, Andreas Meister

Modified Patankar-Runge-Kutta (MPRK) schemes are numerical methods for the solution of positive and conservative production-destruction systems. They adapt explicit Runge-Kutta schemes to ensure positivity and conservation irrespective of the time step size. The first two members of this class were introduced in [Burchard et. al.: A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations. Appl. Numer. Math., 47(1):1-30, 2003] and have been successfully applied in a large number of applications. Recently, we introduced a general definition of MPRK schemes and presented a thorough investigation of first and second order MPRK schemes in [Kopecz. S, Meister, A.: On order conditions for modified patankar-runge-kutta schemes. arXiv:1702.04589 [math.NA], 2017.]. A potentially third order Patankar-type method was introduced in [Formaggia L., Scotti, A.: Positivity and conservation properties of some integration schemes for mass action kinetics. SIAM J. Numer. Anal., 49(3):1267-1288, 2011.]. This method uses the MPRK22(1) scheme of [Burchard et. al.: A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations. Appl. Numer. Math., 47(1):1-30, 2003] as a predictor and a modification of the BDF(3) multistep method as a corrector. This method is at most third order and at least second order accurate. In this paper we continue the work of [Kopecz. S, Meister, A.: On order conditions for modified patankar-runge-kutta schemes. arXiv:1702.04589 [math.NA], 2017.] and present necessary and sufficient conditions for third order MPRK schemes. For the first time, we introduce MPRK schemes, which are third order accurate independent of the specific positive and conservative system under consideration. The theoretical results are confirmed by numerical experiments.