Comptes Rendus
no. 19-20
Partial Differential Equations/Numerical Analysis
Convergence of semi-discrete approximations of Benney equations
[Convergence d'une approximation semi-discrète des équations de Benney]

Présenté par : Gilles Lebeau

Paulo Amorim1 ; Mário Figueira1
1 Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1135-1140.

Dans la première partie de cette Note, on étudie l'approximation numérique des équations de Benney dans le cas de résonance des ondes courtes et longues. On prouve la convergence d'un schéma aux différences finies semi-discret dans l'espace de l'énérgie. Dans la deuxième partie de cette Note, on condidère une version quasilinéaire des équations de Benney. On prouve la convergence d'un schéma du type Lax–Friedrichs semi-discret vers la solution d'entropie du problème.

In the first part of this Note we study the numerical approximation of Benney equations in the long wave-short wave resonance case. We prove the convergence of a finite-difference semi-discrete scheme in the energy space. In the second part of the Note we consider the semi-discretization of a quasilinear version of Benney equations. We prove the convergence of a finite-difference semi-discrete Lax–Friedrichs type scheme towards a weak entropy solution of the Cauchy problem.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.08.002
Paulo Amorim 1 ; Mário Figueira 1

1 Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal 
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     title = {Convergence of semi-discrete approximations of {Benney} equations},
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Paulo Amorim; Mário Figueira. Convergence of semi-discrete approximations of Benney equations. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1135-1140. doi : 10.1016/j.crma.2009.08.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.08.002/

[1] P. Amorim, M. Figueira, Convergence of numerical schemes for interaction equations of short and long waves, in preparation

[2] D. Bekiranov; T. Ogawa; G. Ponce Interaction equations for short and long dispersive waves, J. Funct. Anal., Volume 158 (1998) no. 2, pp. 357-388

[3] D.J. Benney A general theory for interactions between short and long waves, Stud. Appl. Math., Volume 56 (1977), pp. 81-94

[4] J.-P. Dias, M. Figueira, H. Frid, Vanishing viscosity with short wave long wave interactions for systems of conservation laws, Arch. Ration. Mech. Anal., in press

[5] R.J. DiPerna Convergence of approximate solutions to conservation laws, Arch. Ration. Mech. Anal., Volume 82 (1983) no. 1, pp. 27-70

[6] L.I. Ignat; E. Zuazua Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris, Volume 340 (2005) no. 7, pp. 529-534

[7] F. Murat Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 5 (1978) no. 3, pp. 489-507

[8] L. Tartar Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, vol. IV, Pitman, Boston, 1979, pp. 136-212

[9] M. Tsutsumi; S. Hatano Well-posedness of the Cauchy problem for the long wave–short wave resonance equations, Nonlinear Anal., Volume 22 (1994) no. 2, pp. 155-171

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