In this work we design a new domain decomposition method for the Euler equations in dimensions. The starting point is the equivalence with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation [Achdou and Nataf, C. R. Acad. Sci. Paris Sér. I 325 (1997) 1211-1216]. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into sub-domains, it converges in iterations. This property cannot be conserved strictly at discrete level and for arbitrary domain decompositions but we still have numerical results which confirm a very good stability with respect to the various parameters of the problem (mesh size, Mach number, ).
@article{M2AN_2006__40_4_689_0,
author = {Dolean, Victorita and Nataf, Fr\'ed\'eric},
title = {A new domain decomposition method for the compressible Euler equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {40},
year = {2006},
pages = {689-703},
doi = {10.1051/m2an:2006026},
mrnumber = {2274774},
zbl = {pre05122051},
zbl = {1173.76381},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2006__40_4_689_0}
}
Dolean, Victorita; Nataf, Frédéric. A new domain decomposition method for the compressible Euler equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) pp. 689-703. doi : 10.1051/m2an:2006026. http://gdmltest.u-ga.fr/item/M2AN_2006__40_4_689_0/
[1] and, A Robin-Robin preconditioner for an advection-diffusion problem. C. R. Acad. Sci. Paris Sér. I 325 (1997) 1211-1216. | Zbl 0893.65061
[2] ,, and, A domain decomposition preconditioner for an advection-diffusion problem. Comput. Methods Appl. Mech. Engrg. 184 (2000) 145-170. | Zbl 0979.76043
[3] and, A domain decomposition method for the Helmholtz equation and related optimal control. J. Comp. Phys. 136 (1997) 68-82. | Zbl 0884.65118
[4] , A note on the convergence of discretized dynamic iteration. BIT 35 (1995) 291-296. | Zbl 0833.65074
[5] ,, and, Variational formulation and algorithm for trace operator in domain decomposition calculations, in Domain Decomposition Methods, T. Chan, R. Glowinski, J. Périaux and O. Widlund Eds., Philadelphia, PA, SIAM (1989) 3-16. | Zbl 0684.65094
[6] , and, A minimum overlap restricted additive Schwarz preconditioner and appication in 3D flow simulations, in Proceedings of the 10th Domain Decomposition Methods in Sciences and Engineering, C. Farhat J. Mandel and X.-C. Cai Eds., Contemporary Mathematics, AMS 218 (1998) 479-485. | Zbl 0936.76036
[7] and, Symmetrized method with optimized second-order conditions for the Helmholtz equation, in Domain Decomposition Methods, 10 (Boulder, CO, 1997). Amer. Math. Soc., Providence, RI (1998) 400-407. | Zbl 0909.65105
[8] , Non-overlapping Schwarz method for systems of first order equations. Cont. Math. 218 (1998) 408-416. | Zbl 0935.76049
[9] and, An optimized Schwarz algorithm for the compressible Euler equations. Technical Report 556, CMAP, École Polytechnique (2004).
[10] , and, Construction of interface conditions for solving compressible Euler equations by non-overlapping domain decomposition methods. Int. J. Numer. Meth. Fluids 40 (2002) 1485-1492. | Zbl 1025.76021
[11] , and, Convergence analysis of a Schwarz type domain decomposition method for the solution of the Euler equations. Appl. Num. Math. 49 (2004) 153-186. | Zbl pre02083291
[12] and, Absorbing boundary conditions for domain decomposition. Appl. Numer. Math. 27 (1998) 341-365. | Zbl 0935.65135
[13] and, Méthodes de relaxation d'ondes pour l'équation de la chaleur en dimension 1. C.R. Acad. Sci. Paris, Sér. I 336 (2003) 519-524. | Zbl 1028.65100
[14] , and, Optimal Schwarz waveform relaxation for the one dimensional wave equation. Technical Report 469, CMAP, École Polytechnique (2001). | Zbl 1085.65077
[15] , and, Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 (2002) 38-60. | Zbl 1021.65061
[16] , Théorie des matrices. Tome 1: Théorie générale. Traduit du Russe par C. Sarthou. Collection Universitaire de Mathématiques, No. 18. Dunod, Paris (1966). | MR 225788 | Zbl 0136.00410
[17] , Théorie des matrices. Tome 2: Questions spéciales et applications. Traduit du Russe par C. Sarthou. Collection Universitaire de Mathématiques, No. 19. Dunod, Paris (1966). | MR 225789 | Zbl 0136.00410
[18] , Theorie des matrices. Dunod (1966). | Zbl 0136.00410
[19] , The theory of matrices. Vol. 1. AMS Chelsea Publishing, Providence, RI (1998). Translated from the Russian by K.A. Hirsch, Reprint of the 1959 translation. | MR 1657129 | Zbl 0927.15001
[20] , and, A Robin-Robin preconditioner for advection-diffusion equations with discontinuous coefficients. Comput. Methods Appl. Mech. Engrg. 193 (2004) 745-764. | Zbl 1053.76039
[21] ,,, and, Eds. Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, Philadelphia, PA, SIAM (1991). | MR 1106444 | Zbl 0758.00010
[22] , and, The optimized order 2 method. Application to convection-diffusion problems. Future Generation Computer Systems FUTURE 18 (2001). | Zbl 1050.65124
[23] , and, A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays. J. Comput. Phys. 203 (2005) 1-21. | Zbl 1059.78042
[24] , A Dual-Primal FETI method for incompressible Stokes equations. Numer. Math. 102 (2005) 257-275. | Zbl pre02245459
[25] and, BDDC algorithms for incompressible Stokes equations. Technical report (2006) (submitted). | MR 2272601 | Zbl pre05223840
[26] , On the Schwarz alternating method. III: a variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, March 20-22, 1989, T.F. Chan, R. Glowinski, J. Périaux and O. Widlund, Eds., Philadelphia, PA, SIAM (1990). | Zbl 0704.65090
[27] , Balancing domain decomposition. Commun. Appl. Numer. M. 9 (1992) 233-241. | Zbl 0796.65126
[28] , Domain decomposition methods for systems of conservation laws: spectral collocation approximation. SIAM J. Sci. Stat. Comput. 11 (1990) 1029-1052. | Zbl 0711.65082
[29] and, Homogeneous and heterogeneous domain decomposition methods for compressible flow at high reynolds numbers. Technical Report 33, CRS4 (1996). | Zbl 0771.65061
[30] and, Analysis and Test of a Local Domain Decomposition Preconditioner, in R. Glowinski et al. [21] (1991). | Zbl 0770.65082
[31] and, Domain Decomposition Methods - Algorithms and Theory. Springer Series in Computational Mathematics. Springer Verlag (2004). | Zbl 1069.65138
[32] , and, Boundary value problems for elliptic systems. Cambridge University Press, Cambridge (1995). | MR 1343490 | Zbl 0836.35042