Summary
Robust and efficient algorithms for solving linear hyperbolic and elliptic systems of first order are presented. A vertex-centered discretisation of the conservation equations on a triangular grid is applied. The discrete system is obtained by minimizing the flux residuals on the triangles. The resulting normal equations are solved by a subspace decomposition approach. A new sequence of four prolongations defines the subspaces.
Additive and multiplicative Schwarz iterations modified by smoothing steps solve the equations. Robust relaxation schemes with uniformly bounded error reduction rates for anisotropic elliptic and hyperbolic problems are obtained. The combination of a conjugate gradient method with modified additive Schwarz iteration as preconditioner improves the convergence.
The additive variants are perfectly suited for parallel processing. A parallel efficiency up to 0.8 is obtained with nine processors.
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References
Jameson, A.: Solution of the Euler equations for two dimensional transonic flow by a multigrid method. Applied Math. Computation 13 (1983) 327–356.
Hemker, P. W.; Spekreijse, S. P.: Multiple grid and Osher’s scheme for the efficient solution of the Euler equations. Applied Numer. Math. 2 (1986) 475–493.
Shaw, G.; Wesseling, P.: A multigrid method for the Navier-Stokes equations. Delft University of Technology, Dept. Mathematics and Informatics. Report No.: 86-13, 1986.
Dick, E.: Multigrid formulation of polynomial flux-difference splitting for steady Euler equations. J. Computational Physics 91 (1990) 161–173.
Mulder, W. A.: A high resolution Euler solver based on multigrid, semi-coarsening, and defect correction. J. Computational Physics 100 (1992) 91–104.
Ta’asan, Sholmo: Optimal multigrid solvers for steady state inviscid flow problems. Fourth European Multigrid Conference, EMG’ 93, Amsterdam, 1993.
Koren, P. W.; Hemker, P. W.: Damped, direction-dependent multigrid for hypersonic flow computation. Applied Numerical Mathematics 7 (1991) 309–328.
Hackbusch, W.: A new approach to robust multi-grid solvers. First. Int. Conf. on Industrial and Applied Mathematics. ICI AM’ 87. Proceedings, Philadelphia, PA: SIAM 1988, pp. 114–126.
Hackbusch, W.: The frequency decomposition multi-grid method. Part I: Application to anisotropic equations. Numerische Mathematik 56 (1990) 229–245.
Hughes, Th. J. R.: Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations. Int. J. for Numer. Methods in Fluid Dynamics 7 (1987) 1261–1275.
Hughes, Th. J. R.; Franca, L. P.; Hulbert, G. M.: A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/Least-Squares method for advective-diffusive equations. Computer Methods in Applied Mechanics and Engineering 73 (1989) 173–189.
Hackbusch, W.: Multi-grid methods and applications. Berlin: Springer, 1985.
Katzer, E.: A subspace decomposition twogrid method for hyperbolic equations. Fourth European Conference on Multigrid Methods, EMG’ 93. In: Hemker, P. W.; Wesseling, P. (esd.): Contributions to Multigrid, pp. 87-98. CWI Tract 103. Amsterdam: Centrum voor Wiskunde en Informatica, 1994.
Bertsekas, D. P.; Tsitsiklis, J. N.: Parallel and distributed computation. Englewood Cliffs, NJ: Prentice Hall, 1989.
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© 1995 Springer Fachmedien Wiesbaden
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Katzer, E. (1995). A parallel subspace decomposition method for elliptic and hyperbolic systems. In: Hackbusch, W., Wittum, G. (eds) Fast Solvers for Flow Problems. Notes on Numerical Fluid Mechanics (NNFM). Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14125-9_14
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DOI: https://doi.org/10.1007/978-3-663-14125-9_14
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
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