Davide Torlo, Philipp Öffner, and I have published our new preprint A New Stability Approach for Positivity-Preserving Patankar-type Schemes on arXiv.
Patankar-type schemes are linearly implicit time integration methods designed to be unconditionally positivity-preserving by going outside of the class of general linear methods. Thus, classical stability concepts cannot be applied and there is no satisfying stability theory for these schemes. We develop a new approach to study stability properties of Patankar-type methods. In particular, we demonstrate problematic behavior of these methods that can lead to undesired oscillations or order reduction. Extreme cases of the latter manifest as spurious steady states. We investigate various classes of Patankar-type schemes based on classical Runge-Kutta methods, strong stability preserving Runge-Kutta methods, and deferred correction schemes using our approach. Finally, we strengthen our analysis with challenging applications including stiff nonlinear problems.