Comptes Rendus
no. 3
Partial differential equations/Numerical analysis
Space/time convergence analysis of a class of conservative schemes for linear wave equations

Presented by: Philippe G. Ciarlet

Juliette Chabassier1, 2; Sébastien Imperiale3
1 Magique 3D team – Inria Bordeaux Sud-Ouest, 200, avenue de la Vieille-Tour, 33405 Talence cedex, France
2 Université de Pau et des pays de l'Adour, avenue de l'Université, 64013 Pau cedex, France
3 Inria and Paris-Saclay University, 1, rue Honoré-d'Estienne-d'Orves, 91120 Palaiseau, France
Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 282-289.

This paper concerns the space/time convergence analysis of conservative two-step time discretizations for linear wave equations. Explicit and implicit, second- and fourth-order schemes are considered, while the space discretization is given and satisfies minimal hypotheses. Convergence analysis is done using energy techniques and holds if the time step is upper-bounded by a quantity depending on space discretization parameters. In addition to showing the convergence for recently introduced fourth-order schemes, the novelty of this work consists in the independency of the convergence estimates with respect to the difference between the time step and its greatest admissible value.

Ce travail concerne l'analyse de convergence espace/temps de schémas en temps à deux pas pour les équations d'onde linéaires. Sont considérés des schémas explicites et implicites, d'ordre deux et quatre, tandis que la discrétisation spatiale est donnée et satisfait des hypothèses minimales. L'analyse de convergence est faite par techniques d'énergie et est valide si le pas de temps est borné par une quantité dépendant des paramètres de discrétisation spatiale. En plus de montrer la convergence pour des schémas d'ordre quatre récemments introduits, la nouveauté de ce travail réside dans le fait que les estimations ne dépendent pas de la différence entre le pas de temps et sa plus grande valeur admissible.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.12.009

Juliette Chabassier 1, 2; Sébastien Imperiale 3

1 Magique 3D team – Inria Bordeaux Sud-Ouest, 200, avenue de la Vieille-Tour, 33405 Talence cedex, France
2 Université de Pau et des pays de l'Adour, avenue de l'Université, 64013 Pau cedex, France
3 Inria and Paris-Saclay University, 1, rue Honoré-d'Estienne-d'Orves, 91120 Palaiseau, France 
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     title = {Space/time convergence analysis of a class of conservative schemes for linear wave equations},
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Juliette Chabassier; Sébastien Imperiale. Space/time convergence analysis of a class of conservative schemes for linear wave equations. Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 282-289. doi : 10.1016/j.crma.2016.12.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.12.009/

[1] J. Chabassier; S. Imperiale Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string, Wave Motion, Volume 50 (2012) no. 3, pp. 456-480

[2] J. Chabassier; S. Imperiale Introduction and study of fourth-order theta schemes for linear wave equations, J. Comput. Appl. Math., Volume 245 (2013), pp. 194-212

[3] R. Dautray; J.-L. Lions; R. Dautray; J.-L. Lions Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5: Evolution Problems I, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 6: Evolution Problems II, Springer-Verlag, Berlin, 2000

[4] M. Durufle; P. Grob; P. Joly Influence of Gauss and Gauss–Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain, Numer. Methods Partial Differ. Equ., Volume 25 (2009), pp. 526-551

[5] J.-C. Gilbert; P. Joly Higher order time stepping for second order hyperbolic problems and optimal CFL conditions, Partial Differential Equations, Computational Methods in Applied Sciences, vol. 16, 2008, pp. 67-93

[6] P. Joly Variational methods for time-dependent wave propagation problems, Topics in Computational Wave Propagation, Lecture Notes in Computational Science and Engineering, vol. 31, Springer, Berlin, 2003, pp. 201-264

[7] P. Joly The mathematical model for elastic wave propagation, Effective Computational Methods for Wave Propagation, Numerical Insights, vol. 5, Chapman & Hall/CRC, 2008, pp. 247-266

[8] Y. Maday; A.T. Patera Spectral Element Methods for the Incompressible Navier–Stokes Equations, State-of-the-Art Surveys on Computational Mechanics, American Society of Mechanical Engineers, 1989

[9] G.R. Shubin; J.B. Bell A modified equation approach to constructing fourth-order methods for acoustic wave propagation, SIAM J. Sci. Stat. Comput., Volume 8 (1987) no. 2, pp. 135-151

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