Issues with Positivity-Preserving Patankar-type Schemes

TL;DR

This paper investigates the stability and robustness issues of positivity-preserving Patankar-type schemes, revealing their limitations through theoretical analysis and numerical simulations on linear and nonlinear stiff problems.

Contribution

The paper provides the first detailed analysis of stability problems in Patankar-type schemes, highlighting their potential for oscillations and order reduction even in simple linear cases.

Findings

Patankar schemes can exhibit oscillations on simple linear problems

Order reduction occurs for vanishing initial conditions

Theoretical results extend to nonlinear stiff problems

Abstract

Patankar-type schemes are linearly implicit time integration methods designed to be unconditionally positivity-preserving. However, there are only little results on their stability or robustness. We suggest two approaches to analyze the performance and robustness of these methods. In particular, we demonstrate problematic behaviors of these methods that, even on very simple linear problems, can lead to undesired oscillations and order reduction for vanishing initial condition. Finally, we demonstrate in numerical simulations that our theoretical results for linear problems apply analogously to nonlinear stiff problems.