On the non-global linear stability and spurious fixed points of MPRK schemes with negative RK parameters

TL;DR

This paper investigates the linear stability of modified Patankar--Runge--Kutta schemes, revealing that schemes with negative RK parameters exhibit only local stability, while those with nonnegative parameters can have global stability.

Contribution

The study demonstrates that negative RK parameters lead to only local stability in MPRK schemes, challenging previous assumptions of global stability in such methods.

Findings

Negative RK parameters cause local stability in MPRK schemes.

Nonnegative RK parameters can ensure global linear stability.

Numerical experiments support the conjecture about stability properties.

Abstract

Recently, a stability theory has been developed to study the linear stability of modified Patankar--Runge--Kutta (MPRK) schemes. This stability theory provides sufficient conditions for a fixed point of an MPRK scheme to be stable as well as for the convergence of an MPRK scheme towards the steady state of the corresponding initial value problem, whereas the main assumption is that the initial value is sufficiently close to the steady state. Initially, numerical experiments in several publications indicated that these linear stability properties are not only local, but even global, as is the case for general linear methods. Recently, however, it was discovered that the linear stability of the MPDeC(8) scheme is indeed only local in nature. Our conjecture is that this is a result of negative Runge--Kutta (RK) parameters of MPDeC(8) and that linear stability is indeed global, if the RK parameters are nonnegative. To support this conjecture, we examine the family of MPRK22($\alpha$) methods with negative RK parameters and show that even among these methods there are methods for which the stability properties are only local. However, this local linear stability is not observed for MPRK22($\alpha$) schemes with nonnegative Runge-Kutta parameters.