Abstract
Modified Patankar–Runge–Kutta (MPRK) schemes are modifications of Runge–Kutta schemes, which were developed to guarantee unconditional positivity and conservation, when integrating positive and conservative production-destruction systems. This is achieved by the introduction of Patankar-weights and the choice of the Patankar-weight denominators is of fundamental importance to obtain a certain order of accuracy. Recently, necessary and sufficient conditions for third-order three-stage MPRK schemes, as well as the first family of third-order MPRK schemes were presented. The class of MPRK43 schemes can be interpreted as four-stage methods and the question arises, whether third-order MPRK schemes with only three stages can be constructed. In this paper, we prove that it is impossible to construct third-order MPRK schemes with only three stages, when the usual practice, which takes products of powers of previous stage values as Patankar-weight denominators, is applied.
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Kopecz, S., Meister, A. On the existence of three-stage third-order modified Patankar–Runge–Kutta schemes. Numer Algor 81, 1473–1484 (2019). https://doi.org/10.1007/s11075-019-00680-3
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DOI: https://doi.org/10.1007/s11075-019-00680-3