A Stability Analysis of Modified Patankar-Runge-Kutta methods for a nonlinear Production-Destruction System

TL;DR

This paper analyzes the stability of modified Patankar-Runge-Kutta methods applied to nonlinear production-destruction systems, demonstrating their stability for all time steps through theoretical analysis and numerical experiments.

Contribution

It applies a new stability theorem for non-hyperbolic fixed points to second order MPRK methods, establishing their unconditional stability for nonlinear PDS.

Findings

Fixed points are stable for all time steps.

Second order MPRK methods maintain stability in nonlinear PDS.

Numerical experiments support theoretical stability results.

Abstract

Modified Patankar-Runge-Kutta (MPRK) methods preserve the positivity as well as conservativity of a production-destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong to the class of general linear methods, i.e. the iterates are generated by a nonlinear map $\mathbf g$ even when the PDS is linear. Moreover, due to the conservativity of the method, the map $\mathbf g$ possesses non-hyperbolic fixed points. Recently, a new theorem for the investigation of stability properties of non-hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.