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We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes, Arch. Ration. Mech. Anal. 178 (2005) 373-410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi and R. Camassa, J. Fluid Mech. 313 (1996) 83-103]. We study the well-posedness of such systems, and the asymptotic convergence of their solutions towards solutions of the full Euler system. Then, we provide a rigorous justification of the so-called KdV approximation, stating that any bounded solution of the full Euler system can be decomposed into four propagating waves, each of them being well approximated by the solutions of uncoupled Korteweg-de Vries equations. Our method also applies for models with the rigid lid assumption, using the Boussinesq/Boussinesq models introduced in [J.L. Bona, D. Lannes and J.-C. Saut, J. Math. Pures Appl. 89 (2008) 538-566]. Our explicit and simultaneous decomposition allows to study in details the behavior of the flow depending on the depth and density ratios, for both the rigid lid and free surface configurations. In particular, we consider the influence of the rigid lid assumption on the evolution of the interface, and specify its domain of validity. Finally, solutions of the Boussinesq/Boussinesq systems and the KdV approximation are numerically computed, using a Crank-Nicholson scheme with a predictive step inspired from [C. Besse, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1427-1432; C. Besse and C.H. Bruneau, Math. Mod. Methods Appl. Sci. 8 (1998) 1363-1386].
@article{M2AN_2012__46_1_145_0,
author = {Duch\^ene, Vincent},
title = {Boussinesq/Boussinesq systems for internal waves with a free surface, and the {KdV} approximation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {145--185},
publisher = {EDP-Sciences},
volume = {46},
number = {1},
year = {2012},
doi = {10.1051/m2an/2011037},
mrnumber = {2846370},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011037/}
}
TY - JOUR AU - Duchêne, Vincent TI - Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 145 EP - 185 VL - 46 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011037/ DO - 10.1051/m2an/2011037 LA - en ID - M2AN_2012__46_1_145_0 ER -
%0 Journal Article %A Duchêne, Vincent %T Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 145-185 %V 46 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011037/ %R 10.1051/m2an/2011037 %G en %F M2AN_2012__46_1_145_0
Duchêne, Vincent. Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 1, pp. 145-185. doi: 10.1051/m2an/2011037
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