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

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.