On Order Conditions for modified Patankar-Runge-Kutta schemes

TL;DR

This paper develops a general framework for modified Patankar-Runge-Kutta schemes, establishing conditions for their order of accuracy and introducing two new second-order families that ensure positivity and conservation.

Contribution

It provides a unified definition and necessary conditions for first and second order schemes, along with two novel second-order families of modified Patankar-Runge-Kutta methods.

Findings

Derived necessary and sufficient conditions for scheme order

Introduced two new second-order scheme families

Ensured positivity and conservation in numerical solutions

Abstract

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.