×

zbMATH — the first resource for mathematics

Simulation of laser propagation in a plasma with a frequency wave equation. (English) Zbl 1132.76040
Summary: We perform numerical simulations of the propagation of a laser beam in plasma. At each time step, one has to solve Helmholtz equation in a domain which consists of some hundreds of millions of cells. To solve this huge linear system, we use an iterative Krylov method preconditioned by a separable matrix. The corresponding linear system is solved with a block cyclic reduction method. Some enlightenments on the parallel implementation are also given. Lastly, numerical results are presented including some features concerning the scalability of the numerical method on a parallel architecture.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
65F10 Iterative numerical methods for linear systems
65Y05 Parallel numerical computation
Software:
HERA
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ballereau, Ph.; Casanova, M.; Duboc, F.; Dureau, D.; Jourdren, H.; Loiseau, P.; Metral, J.; Morice, O.; Sentis, R., Simulation of the paraxial laser propagation coupled with hydrodynamics in 3D geometry, J. sci. comput., 33, 1-24, (2007) · Zbl 1177.82104
[2] Benamou, J.D.; Després, B., A domain decomposition method for the Helmholtz equation and related optimal control, J. comput. phys., 136, 68-82, (1997) · Zbl 0884.65118
[3] Berenger, Jean-Pierre, A perfectly matched layer for the absorption of electromagnetic waves, J. comput. phys., 114, 2, 185-200, (1994) · Zbl 0814.65129
[4] Dhillon, Inderjit S.; Parlett, Beresford N., Orthogonal eigenvectors and relative gaps, SIAM J. matrix anal. appl., 25, 3, 858-899, (2003) · Zbl 1068.65046
[5] Dorr, R.M.; Garaizar, X.; Hittinger, J.A.F., Simulation of laser-plasma filamentation using adaptive mesh refinement, J. comput. phys., 177, 233-263, (2002) · Zbl 1045.76024
[6] Doumic, Marie; Golse, François; Sentis, Rémi, Un modèle paraxial de propagation de la lumière: problème aux limites pour l’équation d’advection Schrödinger en coordonnées obliques, C.R. math. acad. sci. Paris, 336, 1, 23-28, (2003) · Zbl 1038.35132
[7] Erlangga, Y.A.; Vuik, C.; Oosterlee, C.W., Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation, Appl. numer. math., 56, 5, 648-666, (2006) · Zbl 1094.65041
[8] Sulem, P.L.; Papanicolaou, G.C.; Sulem, C.; Wang, X.P., Singular solutions of the zaharov equations for Langmuir turbulence, Phys. fluids B, 3, 969-980, (1991)
[9] Greenbaum, Anne, Iterative methods for solving linear systems, () · Zbl 0883.65022
[10] Heller, Don, Some aspects of the cyclic reduction algorithm for block tridiagonal linear systems, SIAM J. num. anal., 13, 4, 484-496, (1976) · Zbl 0347.65019
[11] Horn, Roger A.; Johnson, Charles R., Matrix analysis, (1986), Cambridge University Press New York, NY, USA · Zbl 1267.15001
[12] Hüller, S.; Mounaix, Ph.; Pesme, D.; Tikhonchuk, V.T., Interaction of two neighboring laser beams, Phys. plasmas, 4, 2670-2680, (1997)
[13] Sentis, R.; Benamou, J.D.; Lafitte, O.; Solliec, I., A geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics, part ii, the energy, J. comput. appl. math., 167, 91-134, (2004) · Zbl 1054.78003
[14] Jourdren, H., Hera hydrodynamics AMR plateform for multiphysics simulation, ()
[15] Merle, F.; Glangetas, L., Existence of self-similar bow-up solutions for zaharov equation, Commun. math. phys., 160, 173-215, (1994) · Zbl 0808.35137
[16] Pierre-Louis Lions, On the Schwarz alternating method. III: a variant for nonoverlapping subdomains, in: Tony F. Chan, Roland Glowinski, Jacques Périaux, Olof Widlund (Eds.), Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, March 20-22, 1989, Philadelphia, PA, 1990, SIAM.
[17] Meurant, Gérard, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. matrix anal. appl., 13, 3, 707-728, (1992) · Zbl 0754.65029
[18] Parlett, Beresford N., The new QD algorithms, (), 459-491 · Zbl 0835.65059
[19] Plessix, R.E.; Mulder, W.A., Separation-of-variables as a preconditioner for an iterative Helmholtz solver, Appl. numer. math., 44, 3, 385-400, (2003) · Zbl 1013.65117
[20] Rossi, Tuomo; Toivanen, Jari, A parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension, SIAM J. sci. comput., 20, 5, 1778-1796, (1999), electronic · Zbl 0931.65020
[21] Rutishauser, Heinz, Solution of eigenvalue problems with the LR-transformation, Nat. bur. standards appl. math. ser., 1958, 49, 47-81, (1958) · Zbl 0123.11303
[22] Saad, Yousef, Iterative methods for sparse linear systems, (2003), Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 1031.65046
[23] Sentis, Rémi, Mathematical models for laser – plasma interaction, Math. model. numer. anal. (M2AN), 39, 275-318, (2005) · Zbl 1080.35157
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.