On the Stability of Modified Patankar Methods

Patankar schemes have attracted more and more interests as a time-integration method in the last years due to their unconditionally positivity preserving property. Even though they have been become of major interest, it was long time not clear what the stability properties of such schemes are and how they really perform in practice. Recently, a new stability approach has been proposed, based on Lyapnuov stability with an extension of the central manifold theorem, to investigate the stability properties of positive preserving time-integrators. In this paper, we investigate the stability properties of the classical modified Patankar--Runge--Kutta schemes (MPRK) and the modified Patankar Deferred Correction (MPDeC) approaches. We prove that most of the Patankar schemes are stable and we verify our theoretical results with numerical simulations.