Volume 57, Number 4, July-August 2023
|Page(s)||2319 - 2348|
|Published online||11 July 2023|
A study of the local dynamics of modified Patankar DeC and higher order modified Patankar–RK methods
Department of Mathematics, University of Kassel, Kassel, Germany
2 Institute of Mathematics, Johannes Gutenberg University Mainz, Mainz, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 6 June 2023
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.
Mathematics Subject Classification: 65L05 / 65L06 / 65L20
Key words: Lyapunov stability / Conservative / Positivity-preserving / Modified Patankar schemes / Runge–Kutta / Deferred correction
© The authors. Published by EDP Sciences, SMAI 2023
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