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Extension of modified Patankar-Runge-Kutta schemes to nonautonomous production-destruction systems based on Oliver’s approach. (English) Zbl 1459.65110
Summary: The mathematical modeling of various real life applications leads to systems of ordinary differential equations which include crucial properties like the positivity of the solution as well as the conservation of mass or energy. Based on the fundamental work of H. Burchard et al. [Appl. Numer. Math. 47, No. 1, 1–30 (2003; Zbl 1028.80008)], unconditionally positive and conservative modified Patankar-Runge-Kutta schemes (MPRK) are available. These methods are highly stable and often outperform standard Runge-Kutta schemes.
In this article, we extend MPRK methods, named MPRKO methods, using Oliver’s approach [J. Oliver, Math. Comput. 29, 1032–1036 (1975; Zbl 0331.65044)] to improve the accuracy of these schemes in the field of nonautonomous systems. The approach does not demand \(\mathbf{A} \mathbf{e} = \mathbf{c}\) in the Butcher tableau \(( \mathbf{A} , \mathbf{b} , \mathbf{c} )\), where \(\mathbf{e} = ( 1 , \dots , 1 )^T\). Following the general analysis of MPRK schemes described in [S. Kopecz and A. Meister, BIT 58, No. 3, 691–728 (2018; Zbl 1397.65102)], positivity and mass conservation fundamental properties are proven and even conditions concerning the Patankar weights are given to get second order accuracy of the MPRKO methods. Finally, we consider different linear models and a non-linear epidemiological SEIR problem to confirm the theoretical results and to give reliable statements about the accuracy of the novel class of MPRKO methods.
MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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