Issues with positivity-preserving Patankar-type schemes

Davide Torlo; Philipp Öffner; Hendrik Ranocha

doi:10.1016/j.apnum.2022.07.014

Issues with positivity-preserving Patankar-type schemes

Davide Torlo, Philipp Öffner, Hendrik Ranocha

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

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.

Original language	English (US)
Pages (from-to)	117-147
Number of pages	31
Journal	Applied Numerical Mathematics
Volume	182
DOIs	https://doi.org/10.1016/j.apnum.2022.07.014
State	Published - Aug 17 2022
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-09-14
Acknowledgements: D. T. was funded by Team CARDAMOM in Inria–Bordeaux Sud–Ouest, France and by a SISSA Mathematical Fellowship, Italy. P.Ö. gratefully acknowledge support of the Gutenberg Research College, JGU Mainz and the UZH Postdoc Scholarship (Number FK-19-104). H. R. was supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under Germany's Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure. We would like to thank Stefan Kopecz and David Ketcheson for fruitful discussion at the beginning of this project. This project has started with the visit by H.R. in Zurich in 2019 which was supported by the SNF project (Number 175784) and the King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics
Numerical Analysis

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10.1016/j.apnum.2022.07.014

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title = "Issues with positivity-preserving Patankar-type schemes",

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.",

author = "Davide Torlo and Philipp {\"O}ffner and Hendrik Ranocha",

note = "KAUST Repository Item: Exported on 2022-09-14 Acknowledgements: D. T. was funded by Team CARDAMOM in Inria–Bordeaux Sud–Ouest, France and by a SISSA Mathematical Fellowship, Italy. P.{\"O}. gratefully acknowledge support of the Gutenberg Research College, JGU Mainz and the UZH Postdoc Scholarship (Number FK-19-104). H. R. was supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under Germany's Excellence Strategy EXC 2044-390685587, Mathematics M{\"u}nster: Dynamics-Geometry-Structure. We would like to thank Stefan Kopecz and David Ketcheson for fruitful discussion at the beginning of this project. This project has started with the visit by H.R. in Zurich in 2019 which was supported by the SNF project (Number 175784) and the King Abdullah University of Science and Technology (KAUST). This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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doi = "10.1016/j.apnum.2022.07.014",

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AU - Öffner, Philipp

AU - Ranocha, Hendrik

N1 - KAUST Repository Item: Exported on 2022-09-14 Acknowledgements: D. T. was funded by Team CARDAMOM in Inria–Bordeaux Sud–Ouest, France and by a SISSA Mathematical Fellowship, Italy. P.Ö. gratefully acknowledge support of the Gutenberg Research College, JGU Mainz and the UZH Postdoc Scholarship (Number FK-19-104). H. R. was supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under Germany's Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure. We would like to thank Stefan Kopecz and David Ketcheson for fruitful discussion at the beginning of this project. This project has started with the visit by H.R. in Zurich in 2019 which was supported by the SNF project (Number 175784) and the King Abdullah University of Science and Technology (KAUST). This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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N2 - 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.

AB - 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.

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JF - Applied Numerical Mathematics

ER -

Issues with positivity-preserving Patankar-type schemes

Abstract

Bibliographical note

ASJC Scopus subject areas

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