A new domain decomposition method for the compressible Euler equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 4, p. 689-703
In this work we design a new domain decomposition method for the Euler equations in 2 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 2 sub-domains, it converges in 2 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, ...).
DOI : https://doi.org/10.1051/m2an:2006026
Classification:  35M20,  65M55
Keywords: Smith factorization, domain decomposition method, Euler equations
@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},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {4},
     year = {2006},
     pages = {689-703},
     doi = {10.1051/m2an:2006026},
     zbl = {1173.76381},
     zbl = {pre05122051},
     mrnumber = {2274774},
     language = {en},
     url = {http://www.numdam.org/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, Volume 40 (2006) no. 4, pp. 689-703. doi : 10.1051/m2an:2006026. http://www.numdam.org/item/M2AN_2006__40_4_689_0/

[1] Y. Achdou and F. Nataf, A Robin-Robin preconditioner for an advection-diffusion problem. C. R. Acad. Sci. Paris Sér. I 325 (1997) 1211-1216. | Zbl 0893.65061

[2] Y. Achdou, P. Le Tallec, F. Nataf and M. Vidrascu, A domain decomposition preconditioner for an advection-diffusion problem. Comput. Methods Appl. Mech. Engrg. 184 (2000) 145-170. | Zbl 0979.76043

[3] J.D. Benamou and B. Després, A domain decomposition method for the Helmholtz equation and related optimal control. J. Comp. Phys. 136 (1997) 68-82. | Zbl 0884.65118

[4] M. Bjørhus, A note on the convergence of discretized dynamic iteration. BIT 35 (1995) 291-296. | Zbl 0833.65074

[5] J.-F. Bourgat, R. Glowinski, P. Le Tallec and M. Vidrascu, 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] X.-C. Cai, C. Farhat and M. Sarkis, 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] P. Chevalier and F. Nataf, 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] S. Clerc, Non-overlapping Schwarz method for systems of first order equations. Cont. Math. 218 (1998) 408-416. | Zbl 0935.76049

[9] V. Dolean and F. Nataf, An optimized Schwarz algorithm for the compressible Euler equations. Technical Report 556, CMAP, École Polytechnique (2004).

[10] V. Dolean, S. Lanteri and F. Nataf, 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] V. Dolean, S. Lanteri and F. Nataf, 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] B. Engquist and H.-K. Zhao, Absorbing boundary conditions for domain decomposition. Appl. Numer. Math. 27 (1998) 341-365. | Zbl 0935.65135

[13] M.J. Gander and L. Halpern, 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] M.J. Gander, L. Halpern and F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation. Technical Report 469, CMAP, École Polytechnique (2001). | Zbl 1085.65077

[15] M.J. Gander, F. Magoulès and F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 (2002) 38-60. | Zbl 1021.65061

[16] F.R. Gantmacher, 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] F.R. Gantmacher, 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] F.R. Gantmacher, Theorie des matrices. Dunod (1966). | Zbl 0136.00410

[19] F.R. Gantmacher, 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] L. Gerardo-Giorda, P. Le Tallec and F. Nataf, A Robin-Robin preconditioner for advection-diffusion equations with discontinuous coefficients. Comput. Methods Appl. Mech. Engrg. 193 (2004) 745-764. | Zbl 1053.76039

[21] R. Glowinski, Y.A. Kuznetsov, G. Meurant, J. Periaux and O.B. Widlund, Eds. Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, Philadelphia, PA, SIAM (1991). | MR 1106444 | Zbl 0758.00010

[22] C. Japhet, F. Nataf and F. Rogier, The optimized order 2 method. Application to convection-diffusion problems. Future Generation Computer Systems FUTURE 18 (2001). | Zbl 1050.65124

[23] S.-C. Lee, M.N. Vouvakis and J.-F. Lee, 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] J. Li, A Dual-Primal FETI method for incompressible Stokes equations. Numer. Math. 102 (2005) 257-275. | Zbl pre02245459

[25] J. Li and O. Widlund, BDDC algorithms for incompressible Stokes equations. Technical report (2006) (submitted). | MR 2272601 | Zbl pre05223840

[26] P.-L. Lions, 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] J. Mandel, Balancing domain decomposition. Commun. Appl. Numer. M. 9 (1992) 233-241. | Zbl 0796.65126

[28] A. Quarteroni, Domain decomposition methods for systems of conservation laws: spectral collocation approximation. SIAM J. Sci. Stat. Comput. 11 (1990) 1029-1052. | Zbl 0711.65082

[29] A. Quarteroni and L. Stolcis, Homogeneous and heterogeneous domain decomposition methods for compressible flow at high reynolds numbers. Technical Report 33, CRS4 (1996). | Zbl 0771.65061

[30] Y.H. De Roeck and P. Le Tallec, Analysis and Test of a Local Domain Decomposition Preconditioner, in R. Glowinski et al. [21] (1991). | Zbl 0770.65082

[31] A. Toselli and O. Widlund, Domain Decomposition Methods - Algorithms and Theory. Springer Series in Computational Mathematics. Springer Verlag (2004). | Zbl 1069.65138

[32] J. T. Wloka, B. Rowley and B. Lawruk, Boundary value problems for elliptic systems. Cambridge University Press, Cambridge (1995). | MR 1343490 | Zbl 0836.35042