Avec cedram.org  english version
Rechercher un article
Table des matières de ce volume | Article précédent
Eric Blayo; Antoine Rousseau; Manel Tayachi
Boundary conditions and Schwarz waveform relaxation method for linear viscous Shallow Water equations in hydrodynamics
SMAI-Journal of computational mathematics, 3 (2017), p. 117-137
Article PDF
Class. Math.: 65M55
Mots clés: Schwarz waveform relaxation, shallow water equations, domain decomposition, absorbing operators


We propose in the present work an extension of the Schwarz waveform relaxation method to the case of viscous shallow water system with advection term. We first show the difficulties that arise when approximating the Dirichlet to Neumann operators if we consider an asymptotic analysis based on large Reynolds number regime and a small domain aspect ratio. Therefore we focus on the design of a Schwarz algorithm with Robin like boundary conditions. We prove the well-posedness and the convergence of the algorithm.


[1] E. Audusse, P. Dreyfuss & B. Merlet, “Schwarz wave form relaxation for primitive equations of the ocean”, SIAM J. Sci. Comput. 32 (201) no. 5, p. 2908-2936  MR 2729445
[2] E. Blayo, D. Cherel & A. Rousseau, “Towards optimized Schwarz methods for the Navier-Stokes equations”, J. Sci. Comput. 66 (2016), p. 275-295
[3] B. Engquist & A. Majda, “Absorbing boundary conditions for the numerical simulation of waves”, Math. Comput. 31 (1977), p. 245-267
[4] M. J. Gander, “Optimized Schwarz methods”, SIAM Journal on Numerical Analysis 44 (2006) no. 2, p. 699-731
[5] M. J. Gander & L. Halpern, “Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems”, SIAM Journal on Numerical Analysis 45 (2007) no. 2, p. 666-697
[6] M.J. Gander, “Schwarz methods over the course of time”, Electron. Trans. Numer. Anal. 31 (2008), p. 228-255
[7] M.J. Gander, L Halpern & F. Nataf, Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation, in ddm.org, éd., Eleventh International Conference on Domain Decomposition Methods (London, 1998), 1999, p. 27-36
[8] J.-F. Gerbeau & B. Perthame, “Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numerical Validation”, Discrete and Continuous Dynamical Systems - Series B 1 (2001) no. 1, p. 89-102
[9] L. Halpern, “Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems”, SIAM J. Math. Anal. 22 (1991) no. 5, p. 1256-1283
[10] C. Japhet & F. Nataf, The best interface conditions for domain decomposition methods: absorbing boundary conditions, in L. Tourrette, L. Halpern, éd., Absorbing Boundaries and Layers, Domain Decomposition Methods. Applications to Large Scale Computations, Nova Science Publishers, 2003, p. 348–373
[11] P.L. Lions, On the Schwarz alternating method. III. A variant for nonoverlapping subdomains, in SIAM, éd., Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, 1990, p. 202-223
[12] V. Martin, “An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions”, Computers & Fluids 33 (2004) no. 5–6, p. 829 -837, Applied Mathematics for Industrial Flow Problems
[13] V. Martin, “Schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations”, SIAM J. Sci. Comput. 31 (2009) no. 5, p. 3595-3625
[14] L. Müller & G. Lube, “A nonoverlapping domain decomposition method for the nonstationary Navier-Stokes problem”, ZAMM J. Appl. Math. Mech. 81 (2001), p. 725-726
[15] F.C. Otto & G. Lube, “A nonoverlapping domain decomposition method for the Oseen equations”, Math. Models Methods Appl. Sci. 8 (1998), p. 1091-1117
[16] L.F. Pavarino & O.B. Widlund, “Balancing Neumann-Neumann methods for incompressible Stokes equations”, Comm. Pure Appl. Math. 55 (2002), p. 302-335
[17] Alfio Quarteroni & Alberto Valli, Domain decomposition methods for partial differential equations, Numerical mathematics and scientific computation, Clarendon Press, Oxford, New York, 1999
[18] J.C. Strikwerda & C.D. Scarbnick, “A domain decomposition method for incompressible flow”, SIAM J. Sci. Comput. 14 (1993), p. 49-67
[19] Linda Sundbye, “Global Existence for the Cauchy Problem for the Viscous Shallow Water Equations”, Rocky Mountain J. Math. 28 (1998) no. 3, p. 1135-1152 Article
[20] M. Tayachi, Couplage de modèles de dimensions hétérogènes et application en hydrodynamique, Ph. D. Thesis, University of Grenoble, 2013
[21] M. Tayachi, A. Rousseau, E. Blayo, N. Goutal & V. Martin, “Design and analysis of a Schwarz coupling method for a dimensionally heterogeneous problem”, Int. J. Num. Meth. Fluids 75 (2014), p. 446-465
[22] A. Toselli & O. Widlund, Domain decomposition methods - Algorithms and theory, Springer, Berlin-Heidelberg, 2005
[23] X. Xu, C.O. Chow & Lui S.H., “On non overlapping domain decomposition methods for the incompressible Navier-Stokes equations”, ESAIM Math. Mod. Num. Anal. 39 (2005), p. 1251-1269