Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Yves Achdou, Patric Le Tallec, Frédéric Nataf, and Marina Vidrascu. A domain decomposition preconditioner for an advection-diffusion problem. Comput. Methods Appl. Mech. Engrg., 184:145–170, 2000.
J.D. Benamou and B. Després. A domain decomposition method for the Helmholtz equation and related optimal control. J. Comp. Phys., 136:68–82, 1997.
X.-C. Cai, C. Farhat, and M. Sarkis. A minimum overlap restricted additive Schwarz preconditioner and applications to 3D flow simulations. Contemporary Mathematics, 218:479–485, 1998.
S. Clerc. Non-overlapping Schwarz method for systems of first order equations. Cont. Math, 218:408–416, 1998.
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.
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., 49:153–186, 2004.
V. Dolean and F. Nataf. A new domain decomposition method for the compressible euler equations. Technical Report 567, CMAP-Ecole Polytechnique, 2004.
V. Dolean and F. Nataf. An optimized schwarz algorithm for the compressible euler equations. Technical Report 556, CMAP-Ecole Polytechnique, 2004.
Bjorn Engquist and Hong-Kai Zhao. Absorbing boundary conditions for domain decomposition. Appl. Numer. Math., 27(4):341–365, 1998.
Felix R. Gantmacher. Theorie des matrices. Dunod, 1966.
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., 193:745–764, 2004.
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.
[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.
M.-J. Gander, F. Magoulès, and F. Nataf. Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput., 24-1:38–60, 2002.
Caroline Japhet, Frédéric Nataf, and Francois Rogier. The optimized order 2 method. application to convection-diffusion problems. Future Generation Computer Systems FUTURE, 18, 2001.
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.
A. Quarteroni and L. Stolcis. Homogeneous and heterogeneous domain decomposition methods for compressible flow at high Reynolds numbers. Technical Report 33, CRS4, 1996.
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.
A. Toselli and O. Widlund. Domain Decomposition Methods — Algorithms and Theory. Springer Series in Computational Mathematics. Springer Verlag, 2004.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Dolean, V., Nataf, F. (2006). Domain Decomposition Algorithms for the Compressible Euler Equations. In: Calgaro, C., Coulombel, JF., Goudon, T. (eds) Analysis and Simulation of Fluid Dynamics. Advances in Mathematical Fluid Mechanics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7742-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7742-7_5
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7741-0
Online ISBN: 978-3-7643-7742-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)