Boundary conditions and Schwarz waveform relaxation method for linear viscous Shallow Water equations in hydrodynamics
Eric Blayo; Antoine Rousseau; Manel Tayachi
The SMAI journal of computational mathematics, Volume 3  (2017), p. 117-137

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.

Published online : 2017-09-15
Classification:  65M55
Keywords: Schwarz waveform relaxation, shallow water equations, domain decomposition, absorbing operators
     author = {Eric Blayo and Antoine Rousseau and Manel Tayachi},
     title = {Boundary conditions and Schwarz waveform relaxation method for linear viscous Shallow Water equations in hydrodynamics},
     journal = {The SMAI journal of computational mathematics},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {3},
     year = {2017},
     pages = {117-137},
     doi = {10.5802/smai-jcm.22},
     language = {en},
     url = {}
Blayo, Eric; Rousseau, Antoine; Tayachi, Manel. Boundary conditions and Schwarz waveform relaxation method for linear viscous Shallow Water equations in hydrodynamics. The SMAI journal of computational mathematics, Volume 3 (2017) , pp. 117-137. doi : 10.5802/smai-jcm.22.

[1] E. Audusse; P. Dreyfuss; B. Merlet Schwarz wave form relaxation for primitive equations of the ocean, SIAM J. Sci. Comput., Volume 32 (201) no. 5, pp. 2908-2936 | Article

[2] E. Blayo; D. Cherel; A. Rousseau Towards optimized Schwarz methods for the Navier-Stokes equations, J. Sci. Comput., Volume 66 (2016), pp. 275-295 | Article

[3] B. Engquist; A. Majda Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., Volume 31 (1977), pp. 245-267 | Article

[4] M. J. Gander Optimized Schwarz methods, SIAM Journal on Numerical Analysis, Volume 44 (2006) no. 2, pp. 699-731 | Article

[5] M. J. Gander; L. Halpern Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems, SIAM Journal on Numerical Analysis, Volume 45 (2007) no. 2, pp. 666-697 | Article

[6] M.J. Gander Schwarz methods over the course of time, Electron. Trans. Numer. Anal., Volume 31 (2008), pp. 228-255

[7] M.J. Gander; L Halpern; F. Nataf; Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation, Eleventh International Conference on Domain Decomposition Methods (London, 1998), Augsburg (1999), pp. 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, Volume 1 (2001) no. 1, pp. 89-102 | Article

[9] L. Halpern Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems, SIAM J. Math. Anal., Volume 22 (1991) no. 5, pp. 1256-1283 | Article

[10] C. Japhet; F. Nataf; L. Tourrette; L. Halpern The best interface conditions for domain decomposition methods: absorbing boundary conditions, Absorbing Boundaries and Layers, Domain Decomposition Methods. Applications to Large Scale Computations, Nova Science Publishers (2003), pp. 348-373

[11] P.L. Lions; SIAM On the Schwarz alternating method. III. A variant for nonoverlapping subdomains, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, Houston (1990), pp. 202-223

[12] V. Martin An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions, Computers & Fluids, Volume 33 (2004) no. 5–6, pp. 829 -837 (Applied Mathematics for Industrial Flow Problems) | Article

[13] V. Martin Schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations, SIAM J. Sci. Comput., Volume 31 (2009) no. 5, pp. 3595-3625 | Article

[14] L. Müller; G. Lube A nonoverlapping domain decomposition method for the nonstationary Navier-Stokes problem, ZAMM J. Appl. Math. Mech., Volume 81 (2001), p. 725-726 | Article

[15] F.C. Otto; G. Lube A nonoverlapping domain decomposition method for the Oseen equations, Math. Models Methods Appl. Sci., Volume 8 (1998), pp. 1091-1117 | Article

[16] L.F. Pavarino; O.B. Widlund Balancing Neumann-Neumann methods for incompressible Stokes equations, Comm. Pure Appl. Math., Volume 55 (2002), pp. 302-335 | Article

[17] Alfio Quarteroni; Alberto Valli Domain decomposition methods for partial differential equations, Clarendon Press, Oxford, New York, Numerical mathematics and scientific computation (1999)

[18] J.C. Strikwerda; C.D. Scarbnick A domain decomposition method for incompressible flow, SIAM J. Sci. Comput., Volume 14 (1993), pp. 49-67 | Article

[19] Linda Sundbye Global Existence for the Cauchy Problem for the Viscous Shallow Water Equations, Rocky Mountain J. Math., Volume 28 (1998) no. 3, pp. 1135-1152 | Article

[20] M. Tayachi Couplage de modèles de dimensions hétérogènes et application en hydrodynamique, University of Grenoble (2013) (Ph. D. Thesis)

[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, Volume 75 (2014), pp. 446-465 | Article

[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., Volume 39 (2005), pp. 1251-1269 | Article