Date
2023-03-24Subject
510 Mathematics Runge-Kutta-VerfahrenDifferentialgleichungStabilitätPhysikalische EigenschaftAnalyseMetadata
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Aufsatz
A Stability Analysis of Modified Patankar–Runge–Kutta methods for a nonlinear Production–Destruction System
Abstract
Modified Patankar–Runge–Kutta (MPRK) methods preserve the positivity as well as conservativity of a production–destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong to the class of general linear methods, i. e. the iterates are generated by a nonlinear map g even when the PDS is linear. Moreover, due to the conservativity of the method, the map g possesses non-hyperbolic fixed points.
Recently, a new theorem for the investigation of stability properties of non-hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.
Recently, a new theorem for the investigation of stability properties of non-hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.
Citation
In: Proceedings in Applied Mathematics and Mechanics (PAMM) Volume 22 / Issue 1 (2023-03-24) eissn:1617-7061Sponsorship
Gefördert im Rahmen des Projekts DEALCitation
@article{doi:10.17170/kobra-202304057781,
author={Izgin, Thomas and Kopecz, Stefan and Meister, Andreas},
title={A Stability Analysis of Modified Patankar–Runge–Kutta methods for a nonlinear Production–Destruction System},
journal={Proceedings in Applied Mathematics and Mechanics (PAMM)},
year={2023}
}
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2023-04-24T09:57:36Z 2023-04-24T09:57:36Z 2023-03-24 doi:10.17170/kobra-202304057781 http://hdl.handle.net/123456789/14624 Gefördert im Rahmen des Projekts DEAL eng Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ 510 A Stability Analysis of Modified Patankar–Runge–Kutta methods for a nonlinear Production–Destruction System Aufsatz Modified Patankar–Runge–Kutta (MPRK) methods preserve the positivity as well as conservativity of a production–destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong to the class of general linear methods, i. e. the iterates are generated by a nonlinear map g even when the PDS is linear. Moreover, due to the conservativity of the method, the map g possesses non-hyperbolic fixed points. Recently, a new theorem for the investigation of stability properties of non-hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims. open access Izgin, Thomas Kopecz, Stefan Meister, Andreas doi:10.1002/pamm.202200083 Runge-Kutta-Verfahren Differentialgleichung Stabilität Physikalische Eigenschaft Analyse publishedVersion eissn:1617-7061 Issue 1 Proceedings in Applied Mathematics and Mechanics (PAMM) Volume 22 false e202200083
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