Domain Decomposition Algorithms for the Compressible Euler Equations
In this work we present an overview of some classical and new domain decomposition methods for the resolution of the Euler equations. The classical Schwarz methods are formulated and analyzed in the framework of first order hyperbolic systems and the differences with respect to the scalar problems are presented. This kind of algorithms behave quite well for bigger Mach numbers but we can further improve their performances in the case of lower Mach numbers. There are two possible ways to achieve this goal. The first one implies the use of the optimized interface conditions depending on a few parameters that generalize the classical ones. The second is inspired from the Robin-Robin preconditioner for the convection-diffusion equation by using the equivalence via the Smith factorization with a third order scalar equation.
KeywordsMach Number Euler Equation Interface Condition Domain Decomposition Method Euler System
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- [CN98]Philippe Chevalier and Frédéric Nataf. Symmetrized method with optimized second-order conditions for the Helmholtz equation. In Domain decomposition methods, 10 (Boulder, CO, 1997), pages 400–407. Amer. Math. Soc., Providence, RI, 1998.Google Scholar
- [DN04a]V. Dolean and F. Nataf. A new domain decomposition method for the compressible euler equations. Technical Report 567, CMAP-Ecole Polytechnique, 2004.Google Scholar
- [DN04b]V. Dolean and F. Nataf. An optimized schwarz algorithm for the compressible euler equations. Technical Report 556, CMAP-Ecole Polytechnique, 2004.Google Scholar
- [Gan66]Felix R. Gantmacher. Theorie des matrices. Dunod, 1966.Google Scholar
- [GHN01]Martin J. Gander, Laurence Halpern, and Frédéric Nataf. Optimal Schwarz waveform relaxation for the one-dimensional wave equation. Technical Report 469, CMAP, Ecole Polytechnique, September 2001.Google Scholar
- [GKM+91]_R. Glowinski, Y.A. Kuznetsov, G. Meurant, J. Periaux, and O.B. Widlund, editors. Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, Philadelphia, 1991. SIAM.Google Scholar
- [JNR01]Caroline Japhet, Frédéric Nataf, and Francois Rogier. The optimized order 2 method. application to convection-diffusion problems. Future Generation Computer Systems FUTURE, 18, 2001.Google Scholar
- [Lio90]Pierre-Louis Lions. On the Schwarz alternating method. III: a variant for nonoverlapping subdomains. In Tony F. Chan, Roland Glowinski, Jacques Périaux, and Olof Widlund, editors, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, March 20–22, 1989, Philadelphia, PA, 1990. SIAM.Google Scholar
- [QS96]A. Quarteroni and L. Stolcis. Homogeneous and heterogeneous domain decomposition methods for compressible flow at high Reynolds numbers. Technical Report 33, CRS4, 1996.Google Scholar
- [RT91]Y.H. De Roeck and P. Le Tallec. Analysis and Test of a Local Domain Decomposition Preconditioner. In Y.A. Kuznetsov, G. Meurant, J. Periaux, and O.B. Widlund, editors. Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, Philadelphia, 1991. SIAM R. Glowinski et al. [GKM+91]}, 1991.Google Scholar
- [TW04]A. Toselli and O. Widlund. Domain Decomposition Methods — Algorithms and Theory. Springer Series in Computational Mathematics. Springer Verlag, 2004.Google Scholar