Geodesic
Stability theory for some scalar finite difference schemes: validity of the modified equations approach
Firas Dhaouadi1 ; Emilie Duval2 ; Sergey Tkachenko3 ; Jean-Paul Vila4
1Université Paul Sabatier, Institut de Mathématiques de Toulouse
2Université Grenoble Alpes, Laboratoire Jean Kuntzmann
3Aix-Marseille Université, CNRS, IUSTI, UMR 7343
4Institut de Mathématiques de Toulouse, INSA Toulouse
ESAIM. Proceedings, Tome 70 (2021), pp. 124-136
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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.
DOI : 10.1051/proc/202107008

Firas Dhaouadi  1   ; Emilie Duval  2   ; Sergey Tkachenko  3   ; Jean-Paul Vila  4

1 Université Paul Sabatier, Institut de Mathématiques de Toulouse
2 Université Grenoble Alpes, Laboratoire Jean Kuntzmann
3 Aix-Marseille Université, CNRS, IUSTI, UMR 7343
4 Institut de Mathématiques de Toulouse, INSA Toulouse 
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     author = {Firas Dhaouadi and Emilie Duval and Sergey Tkachenko and Jean-Paul Vila},
     title = {Stability theory for some scalar finite difference schemes: validity of the modified equations approach},
     journal = {ESAIM. Proceedings},
     pages = {124--136},
     year = {2021},
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     doi = {10.1051/proc/202107008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/202107008/}
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Firas Dhaouadi; Emilie Duval; Sergey Tkachenko; Jean-Paul Vila. Stability theory for some scalar finite difference schemes: validity of the modified equations approach. ESAIM. Proceedings, Tome 70 (2021), pp. 124-136. doi: 10.1051/proc/202107008

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