On Lyapunov Stability of Positive and Conservative Time Integrators and Application to Second Order Modified Patankar--Runge--Kutta Schemes

TL;DR

This paper analyzes the Lyapunov stability of second order modified Patankar--Runge--Kutta schemes, proving unconditional stability for one variant and deriving stability regions for another, supported by numerical experiments.

Contribution

It provides the first analytic stability analysis of MPRK schemes using center manifold theory, establishing stability properties for specific second order methods.

Findings

MPRK22(α) schemes are unconditionally stable.

Stability regions for MPRK22ncs(α) schemes are derived.

Numerical experiments confirm theoretical stability results.

Abstract

Since almost twenty years, modified Patankar--Runge--Kutta (MPRK) methods have proven to be efficient and robust numerical schemes that preserve positivity and conservativity of the production-destruction system irrespectively of the time step size chosen. Due to these advantageous properties they are used for a wide variety of applications. Nevertheless, until now, an analytic investigation of the stability of MPRK schemes is still missing, since the usual approach by means of Dahlquist's equation is not feasible. Therefore, we consider a positive and conservative 2D test problem and provide statements usable for a stability analysis of general positive and conservative time integrator schemes based on the center manifold theory. We use this approach to investigate the Lyapunov stability of the second order MPRK22($\alpha$) and MPRK22ncs($\alpha$) schemes. We prove that MPRK22($\alpha$) schemes are unconditionally stable and derive the stability regions of MPRK22ncs($\alpha$) schemes. Finally, numerical experiments are presented, which confirm the theoretical results.