A study of the local dynamics of modified Patankar DeC and higher order modified Patankar RK methods

TL;DR

This paper analyzes the stability of modified Patankar schemes, demonstrating that most are stable for any time step size and providing theoretical and numerical validation of these stability properties.

Contribution

It introduces a stability analysis for MPRK and MPDeC schemes, showing their unconditional stability and computing the stability function for MPDeC.

Findings

Most MPRK schemes are stable for any time step size.

Most MPDeC schemes are stable irrespective of the time step size.

Numerical simulations confirm the theoretical stability results.

Abstract

Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of the analytical solution of a production-destruction system (PDS) irrespective of the chosen time step size. Although they are now of great interest, for a long time it was not clear what stability properties such schemes have. Recently a new stability approach based on Lyapunov stability with an extension of the center manifold theorem has been proposed to study the stability properties of positivity-preserving time integrators. In this work, we study 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 considered MPRK schemes are stable for any time step size and compute the stability function of MPDeC. We investigate its properties numerically revealing that also most MPDeC are stable irrespective of the chosen time step size. Finally, we verify our theoretical results with numerical simulations.