Stability theory for some scalar finite difference schemes: validity of the modified equations approach*

Stability theory for some scalar finite difference schemes: validity of the modified equations approach*

In this paper, we discuss some limitations of the modified equations approach as a tool for stability analysis for a class of explicit linear schemes to scalar partial differential equations. We show that the infinite series obtained by Fourier transform of the modified equation is not always conver...

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Main Authors: Dhaouadi Firas, Duval Emilie, Tkachenko Sergey, Vila Jean-Paul
Format: Article
Language:English
Published: EDP Sciences 2021-01-01
Series:ESAIM: Proceedings and Surveys
Online Access:https://www.esaim-proc.org/articles/proc/pdf/2021/01/proc2107008.pdf
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spelling doaj-cb7478ef18084f22a6434a6a484dcb1c2021-07-15T14:18:27ZengEDP SciencesESAIM: Proceedings and Surveys2267-30592021-01-017012413610.1051/proc/202107008proc2107008Stability theory for some scalar finite difference schemes: validity of the modified equations approach*Dhaouadi Firas0Duval Emilie1Tkachenko Sergey2Vila Jean-Paul3Université Paul Sabatier, Institut de Mathématiques de ToulouseUniversité Grenoble Alpes, Laboratoire Jean KuntzmannAix-Marseille Université, CNRS, IUSTI, UMR 7343Institut de Mathématiques de Toulouse, INSA ToulouseIn this paper, we discuss some limitations of the modified equations approach as a tool for stability analysis for a class of explicit linear schemes to scalar partial differential equations. We show that the infinite series obtained by Fourier transform of the modified equation is not always convergent and that in the case of divergence, it becomes unrelated to the scheme. Based on these results, we explain when the stability analysis of a given truncation of a modified equation may yield a reasonable estimation of a stability condition for the associated scheme. We illustrate our analysis by some examples of schemes namely for the heat equation and the transport equation.https://www.esaim-proc.org/articles/proc/pdf/2021/01/proc2107008.pdf
collection DOAJ
language English
format Article
sources DOAJ
author Dhaouadi Firas
Duval Emilie
Tkachenko Sergey
Vila Jean-Paul
spellingShingle Dhaouadi Firas
Duval Emilie
Tkachenko Sergey
Vila Jean-Paul
Stability theory for some scalar finite difference schemes: validity of the modified equations approach*
ESAIM: Proceedings and Surveys
author_facet Dhaouadi Firas
Duval Emilie
Tkachenko Sergey
Vila Jean-Paul
author_sort Dhaouadi Firas
title Stability theory for some scalar finite difference schemes: validity of the modified equations approach*
title_short Stability theory for some scalar finite difference schemes: validity of the modified equations approach*
title_full Stability theory for some scalar finite difference schemes: validity of the modified equations approach*
title_fullStr Stability theory for some scalar finite difference schemes: validity of the modified equations approach*
title_full_unstemmed Stability theory for some scalar finite difference schemes: validity of the modified equations approach*
title_sort stability theory for some scalar finite difference schemes: validity of the modified equations approach*
publisher EDP Sciences
series ESAIM: Proceedings and Surveys
issn 2267-3059
publishDate 2021-01-01
description In this paper, we discuss some limitations of the modified equations approach as a tool for stability analysis for a class of explicit linear schemes to scalar partial differential equations. We show that the infinite series obtained by Fourier transform of the modified equation is not always convergent and that in the case of divergence, it becomes unrelated to the scheme. Based on these results, we explain when the stability analysis of a given truncation of a modified equation may yield a reasonable estimation of a stability condition for the associated scheme. We illustrate our analysis by some examples of schemes namely for the heat equation and the transport equation.
url https://www.esaim-proc.org/articles/proc/pdf/2021/01/proc2107008.pdf
work_keys_str_mv AT dhaouadifiras stabilitytheoryforsomescalarfinitedifferenceschemesvalidityofthemodifiedequationsapproach
AT duvalemilie stabilitytheoryforsomescalarfinitedifferenceschemesvalidityofthemodifiedequationsapproach
AT tkachenkosergey stabilitytheoryforsomescalarfinitedifferenceschemesvalidityofthemodifiedequationsapproach
AT vilajeanpaul stabilitytheoryforsomescalarfinitedifferenceschemesvalidityofthemodifiedequationsapproach
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