Geodesic
Stability theory for some scalar finite difference schemes: validity of the modified equations approach
Firas Dhaouadi1 ; Emilie Duval2 ; Sergey Tkachenko3 ; Jean-Paul Vila4
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
ESAIM. Proceedings, Tome 70 (2021), pp. 124-136.

Voir la notice de l'article provenant de la source EDP Sciences

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 
@article{EP_2021_70_a8,
     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},
     publisher = {mathdoc},
     volume = {70},
     year = {2021},
     doi = {10.1051/proc/202107008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/202107008/}
}
TY  - JOUR
AU  - Firas Dhaouadi
AU  - Emilie Duval
AU  - Sergey Tkachenko
AU  - Jean-Paul Vila
TI  - Stability theory for some scalar finite difference schemes: validity of the modified equations approach
JO  - ESAIM. Proceedings
PY  - 2021
SP  - 124
EP  - 136
VL  - 70
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/proc/202107008/
DO  - 10.1051/proc/202107008
LA  - en
ID  - EP_2021_70_a8
ER  - 
%0 Journal Article
%A Firas Dhaouadi
%A Emilie Duval
%A Sergey Tkachenko
%A Jean-Paul Vila
%T Stability theory for some scalar finite difference schemes: validity of the modified equations approach
%J ESAIM. Proceedings
%D 2021
%P 124-136
%V 70
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/proc/202107008/
%R 10.1051/proc/202107008
%G en
%F EP_2021_70_a8
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. http://geodesic.mathdoc.fr/articles/10.1051/proc/202107008/

Cité par Sources :