A new domain decomposition method for the compressible Euler equations

Dolean, Victorita; Nataf, Frédéric

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 4, pp. 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, $...$).

Classification : 35M20, 65M55
Keywords: Smith factorization, domain decomposition method, Euler equations

[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

[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

[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

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

[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

[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

[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

[8] S. Clerc, Non-overlapping Schwarz method for systems of first order equations. Cont. Math. 218 (1998) 408-416. | Zbl

[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

[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.

[12] B. Engquist and H.-K. Zhao, Absorbing boundary conditions for domain decomposition. Appl. Numer. Math. 27 (1998) 341-365. | Zbl

[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

[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

[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

[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). | Zbl | MR

[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). | Zbl | MR

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

[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. | Zbl | MR

[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

[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). | Zbl | MR

[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

[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

[24] J. Li, A Dual-Primal FETI method for incompressible Stokes equations. Numer. Math. 102 (2005) 257-275.

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

[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

[27] J. Mandel, Balancing domain decomposition. Commun. Appl. Numer. M. 9 (1992) 233-241. | Zbl

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

[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

[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

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

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