A necessary condition for non oscillatory and positivity preserving time-integration schemes

TL;DR

This paper explores the relationship between Lyapunov stability and oscillatory behavior in Modified Patankar schemes, establishing a necessary condition to prevent oscillations and verifying it through numerical tests.

Contribution

It introduces a necessary condition linking Lyapunov stability to non-oscillatory behavior in MP schemes, advancing understanding of their stability properties.

Findings

A Lyapunov stability condition is necessary to avoid oscillations.

Theoretical results are validated through numerical experiments.

The work connects stability analysis with oscillation prevention in MP schemes.

Abstract

Modified Patankar (MP) schemes are conservative, linear implicit and unconditionally positivity preserving time-integration schemes constructed for production-destruction systems. For such schemes, a classical stability analysis does not yield any information about the performance. Recently, two different techniques have been proposed to investigate the properties of MP schemes. In Izgin et al. [ESAIM: M2AN, 56 (2022)], inspired from dynamical systems, the Lyapunov stability properties of such schemes have been investigated, while in Torlo et al. [Appl. Numer. Math., 182 (2022)] their oscillatory behaviour has been studied. In this work, we investigate the connection between the oscillatory behaviour and the Lyapunov stability and we prove that a condition on the Lyapunov stability function is necessary to avoid oscillations. We verify our theoretical result on several numerical tests.