Issues with Positivity-Preserving Patankar-type Schemes

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 or robustness theory for these schemes. We develop a new approach to study a few related issues that impact some Patankar-type methods. In particular, we demonstrate problematic behaviors of these methods that can lead to undesired oscillations or order reduction on very simple linear problems. 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.