Abstract
In many astrophysical hydrodynamical and magnetohydrodynamical problems explicit time integration is inefficient due to the disparate time scales, or because we are looking for a steady-state. Implicit schemes can offer a viable alternative, which requires efficient solution of large linear systems. In multi-dimensional flows these linear systems cannot be solved with a direct method, so that iterative methods must be used. Since the physical problems are often advection dominated, the matrix is approximately anti-symmetric and non-diagonally dominant, thus preconditioning is necessary.
We describe a new approach, in which a GMRES type iterative method is combined with an Eisenstat implementation of a relaxed form of a modified Block Incomplete LU-decomposition as preconditioner. Parallelisation aspects of the algorithm are also discussed.
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Aarden, J., Karlsson, K.-E.: Preconditioned cg-type methods for solving the coupled systems of fundamental semiconductor equations. BIT 29 (1989) 916–937
Axelsson, O., Gustafsson, I.: Iterative solution for the solution of the Navier equations of elasticity. Comput. Methods Appl. Mech. Eng. 15 (1978) 241–258
Axelsson, O., G. Lindskog, G.: On the eigenvalue distribution of a class of preconditioning methods. Numer. Math. 48 (1986) 479–498
Dongarra J. J., Duff I. S., Sorensen, D. C., van der Vorst, H. A.: Solving linear systems on vector and shared memory computers. (SIAM, 1991) 174–179
Duff, I. S., Meurant, G. A.: The effect of ordering on preconditioned conjugate gradient. BIT 29 (1989) 635–657
Eisenstat, S. C.: Efficient implementation of a class of preconditioned conjugate gradient methods. SIAM J. Sci. Statist. Comput. 2 (1981) 1–4
Gustafsson, I.: A class of 1:st order factorization methods. BIT 18 (1978) 142–156
Keppens, R., Tóth, G., van der Ploeg, A., Botchev, M.: Implicit and Semi-Implicit Schemes in the Versatile Advection Code. (work in progress)
Saad, Y., Schultz, M. H.: A generalized minimal residual algorithm for solving non-symmetric linear systems. SIAM J. Sci. Statist. Comput. 7 (1986) 856–869
Tóth, G.: A general code for modeling MHD flows on parallel computers: Versatile advection code. Astrophys. Lett. & Comm. 34 (1996), 245–250
Tóth, G.: Versatile Advection Code. (in this volume)
Tóth, G., Odstrčil, D.: Comparison of some Flux Corrected Transport and Total Variation Diminishing Numerical Schemes for Hydrodynamic and Magnetohydrodynamic Problems. J. Comput. Phys. 128 (1996) 82–100
Yee, H. C.: A class of high-resolution explicit and implicit shock-capturing methods. NASA TM-101088 (1989)
van der Ploeg, A.: Preconditioning for sparse matrices with applications. Ph.D. thesis, University of Groningen (1994)
van der Ploeg, A.: Reordering strategies and LU-decomposition of block tridiagonal matrices for parallel processing. Technical Report NM-R9618, Centrum voor Wiskunde en Informatica, October 1996
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© 1997 Springer-Verlag Berlin Heidelberg
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van der Ploeg, A., Keppens, R., Tóth, G. (1997). Block incomplete LU-preconditioners for implicit solution of advection dominated problems. In: Hertzberger, B., Sloot, P. (eds) High-Performance Computing and Networking. HPCN-Europe 1997. Lecture Notes in Computer Science, vol 1225. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0031614
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DOI: https://doi.org/10.1007/BFb0031614
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