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 conservativity of the production-destruction system irrespectively time step size chosen. Due these advantageous properties they are used for a wide variety applications. Nevertheless, until now, an analytic investigation stability MPRK is still missing, since usual approach by means Dahlquist’s equation not feasible. Therefore, we consider positive conservative 2D test problem provide statements usable analysis general integrator based on center manifold theory. We use this investigate Lyapunov second order MPRK22( α ) MPRK22ncs( schemes. prove unconditionally stable derive regions Finally, experiments presented, which confirm theoretical results.