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Journal of Scientific Computing

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04-04-2015

Towards Optimized Schwarz Methods for the Navier–Stokes Equations

Authors: Eric Blayo, David Cherel, Antoine Rousseau

Published in: Journal of Scientific Computing | Issue 1/2016

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Abstract

This paper presents a study of optimized Schwarz domain decomposition methods for Navier–Stokes equations. Once discretized in time, optimal transparent boundary conditions are derived for the resulting Stokes equations, and a series of local approximations for these nonlocal conditions are proposed. Their convergence properties are studied, and numerical simulations are conducted on the test case of the driven cavity with two subdomains. It is shown that conditions involving one or two degrees of freedom can improve the convergence properties of the original algorithm.

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We consider here the non symmetric form of the stress tensor. The alternative consisting in using its symmetric form will be discussed in Sect. 5.

For the sake of readability, the Schwarz indices \(m\) and \(m-1\) have been removed, but \(u^{1}\) should be read as \(u^{1,m}\), and \(u^{2}\) as \(u^{2,m-1}\) as in (23a). In the other case (\(u^{1}=u^{1,m-1}\) and \(u^{2}=u^{2,m}\)), we would obtain the discretized version of (24a).

The time steps from \(t=0\) to \(t=1\) have been previously computed with the monodomain algorithm in order to avoid starting from a zero initial solution, which could hide numerical evidences.

Brakkee, E., Vuik, C., Wesseling, P.: Domain decomposition for the incompressible Navier–Stokes equations: solving subdomain problems accurately and inaccurately. Int. J. Numer. Methods Fluids 26, 1217–1237 (1998)MathSciNetCrossRefMATH

Bruneau, C.H., Saad, Y.: The 2D lid-driven cavity problem revisited. Comput. Fluids 35(3), 23–23 (2006)CrossRef

Dolean, V., Nataf, F., Rapin, G.: Deriving a new domain decomposition method for the Stokes equations using the Smith factorization. Math. Comput. 78, 789–814 (2009)MathSciNetCrossRefMATH

Efendiev, Y., Galvis, J., Lazarov, R., Willems, J.: Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal. 46, 1175–1199 (2012)MathSciNetCrossRefMATH

Engquist, B., Majda, A.: Absorbing boundary conditions for the numerical simulation of waves. Math. Comput. 139, 629–651 (1977)MathSciNetCrossRef

Erturk, E.: Discussions on driven cavity flow. Int. J. Numer. Methods Fluids 60, 275–294 (2009)MathSciNetCrossRefMATH

Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44, 699–731 (2006)MathSciNetCrossRefMATH

Gunzburger, M., Lee, H.: An optimization-based domain decomposition method for the Navier–Stokes equations. SIAM J. Numer. Anal. 37, 1455–1480 (2000)MathSciNetCrossRefMATH

Halpern, L.: Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems. SIAM J. Math. Anal. 22, 1256–1283 (1991)MathSciNetCrossRefMATH

10.

Halpern, L., Schatzman, M.: Artificial boundary conditions for incompressible viscous flows. SIAM J. Math. Anal. 20, 308–353 (1989)MathSciNetCrossRefMATH

11.

Hoang, T.: Space–time domain decomposition methods for mixed formulations of flow and transport problems in porous media. Ph.D. thesis, University of Pierre and Marie Curie (Paris VI) (2013)

12.

Japhet, C., Nataf, F.: The best interface conditions for domain decomposition methods: absorbing boundary conditions. In: Tourrette, L., Halpern, L. (eds.) Absorbing Boundaries and Layers, Domain Decomposition Methods. Applications to Large Scale Computations, pp. 348–373. Nova Science, Hauppauge (2001)

13.

Japhet, C., Nataf, F., Rogier, F.: The optimized order 2 method: application to convection–diffusion problems. Future Gener. Comput. Syst. 18(1), 17–30 (2001)CrossRefMATH

14.

Kong, F., Ma, Y., Lu, J.: An optimization-based domain decomposition method for numerical simulation of the incompressible Navier–Stokes flows. Numer. Methods PDEs 27, 255–276 (2011)MathSciNetCrossRefMATH

15.

Kumar, P.: Purely algebraic domain decomposition methods for the incompressible Navier–Stokes equations. ArXiv e-prints (2011)

16.

Lions, P.L.: On the Schwarz alternating method I. In: Glowinski, R., Golub, G., Meurant, G., Périaux, J. (eds.) Proceedings of the First International Conference on Domain Decomposition Methods, Domain Decomposition Methods for Partial Differential Equations, pp. 1–42. SIAM, Philadelphia (1988)

17.

Lions, P.L.: On the Schwarz alternating method III: a variant for nonoverlapping subdomains. In: Chan, T., Glowinski, R., Périaux, J., Widlund, O. (eds.) Proceedings of the Third International Conference on Domain Decomposition Methods, Domain Decomposition Methods for Partial Differential Equations, pp. 202–223. SIAM, Philadelphia (1990)

18.

Martin, V.: Schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations. SIAM J. Sci. Comput. 31, 3595–3625 (2009)MathSciNetCrossRefMATH

19.

Müller, L., Lube, G.: A nonoverlapping domain decomposition method for the nonstationary Navier–Stokes problem. ZAMM J. Appl. Math. Mech. 81, 725–726 (2001)CrossRefMATH

20.

Nicolaides, R.: Deflation of conjugate gradients with applications to boundary value problems. SIAM J. Numer. Anal. 24, 355–365 (1987)MathSciNetCrossRefMATH

21.

Otto, F., Lube, G.: A nonoverlapping domain decomposition method for the Oseen equations. Math. Models Methods Appl. Sci. 8, 1091–1117 (1998)MathSciNetCrossRefMATH

22.

Pavarino, L., Widlund, O.: Balancing Neumann–Neumann methods for incompressible Stokes equations. Commun. Pure Appl. Math. 55, 302–335 (2002)MathSciNetCrossRefMATH

23.

Ronquist, E.: A domain decomposition solver for the steady Navier–Stokes equations. In: Ilin, A., Scott, L. (eds.) Proceedings of ICOSAHOM-95, pp. 469–485. SIAM, Philadelphia (1996)

24.

Schwarz, H.A.: Über einen Grenzübergang durch alternierendes Verfahren. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 15, 272–286 (1870)

25.

Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: Abstract robust coarse spaces for systems of pdes via generalized eigenproblems in the overlaps. Numer. Math. 126, 741–770 (2014)MathSciNetCrossRefMATH

26.

Strikwerda, J., Scarbnick, C.: A domain decomposition method for incompressible flow. SIAM J. Sci. Comput. 14, 49–67 (1993)MathSciNetCrossRefMATH

27.

Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory. Springer, Berlin (2005)MATH

28.

Willems, J.: Robust multilevel methods for general symmetric positive definite operators. SIAM J. Numer. Anal. 52, 103–124 (2014)MathSciNetCrossRefMATH

29.

Xu, X., Chow, C., Lui, S.H.: On non overlapping domain decomposition methods for the incompressible Navier–Stokes equations. ESAIM Math. Mod. Numer. Anal. 39, 1251–1269 (2005)MathSciNetCrossRefMATH

Title: Towards Optimized Schwarz Methods for the Navier–Stokes Equations
Authors: Eric Blayo
David Cherel
Antoine Rousseau
Publication date: 04-04-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2016
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-015-0020-9

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