Invited Lecture
Stability of Modified Patankar-Runge-Kutta Methods
Institute of Mathematics, University of Kassel, Germany
Abstract
The talk mainly focuses on stability statements of modified Patankar-Runge-Kutta (MPRK) methods, which have proven to be efficient and robust numerical schemes that preserve positivity and conservativity of production-destruction system irrespectively of the time step size chosen. Due to these advantageous properties they are used for a wide variety of applications. Beside a fundamental introduction of MPRK($\alpha$) as well as MPRKncs($\alpha$) methods, the center manifold theory is used in order to investigate the Lyapunov stability of general positive and conservative time integrator schemes. Based on the derived results, we prove that MPRK22($\alpha$) schemes are unconditionally stable and derive the stability regions of MPRK22ncs($\alpha$) methods. Finally, numerical experiments will be presented, which confirm the theoretical results and show the performance of MPRK schemes in comparison with standard Runge-Kutta methods when applied to advection-diffusion-reaction systems.
Joint work with Stefan Kopecz and Thomas Izgin (University of Kassel, Germany)