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L2-stability of the upwind first order finite volume scheme for the Maxwell equations in two and three dimensions on arbitrary unstructured meshes

Published online by Cambridge University Press:  15 April 2002

Serge Piperno
Serge Piperno*
Affiliation:
CERMICS, INRIA, B.P. 93, 06902 Sophia-Antipolis Cedex, France. (Serge.Piperno@sophia.inria.fr)
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Abstract

We investigate sufficient and possibly necessary conditions for the L2 stability of the upwind first order finite volume scheme for Maxwell equations, with metallic and absorbing boundary conditions. We yield a very general sufficient condition, valid for any finite volume partition in two and three space dimensions. We show this condition is necessary for a class of regular meshes in two space dimensions. However, numerical tests show it is not necessary in three space dimensions even on regular meshes. Stability limits for time and space schemes with higher orders of accuracy are numerically investigated.

Keywords

Electromagnetismfinite volume methodsL2 stabilityenergy methodsunstructured meshesabsorbing boundary conditionmetallic boundary condition.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 1 , January 2000 , pp. 139 - 158
Copyright
© EDP Sciences, SMAI, 2000

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References

Ambrosiano, J.J., Brandon, S.T., Löhner, R. and DeVore, C.R., Electromagnetics via the Taylor-Galerkin finite element method on unstructured grids. J. Comput. Phys. 110 (1994) 310-319. CrossRef
D.A. Anderson, J.C. Tannehill and R.H. Pletcher, Computational fluid mechanics and heat transfer, Hemisphere, McGraw-Hill, New York (1984).
F. Bourdel, P.-A. Mazet and P. Helluy, Resolution of the non-stationary or harmonic Maxwell equations by a discontinuous finite element method. Application to an E.M.I. (electromagnetic impulse) case. Computing Methods in Applied Sciences and Engineering. Nova Science Publishers, Inc., New-York (1991) 405-422.
P.G. Ciarlet and J.-L. Lions Eds., Handbook of Numerical Analysis, Vol. 1. North Holland-Elsevier Science Publishers, Amsterdam, New York, Oxford (1991).
J.-P. Cioni, L. Fezoui and H. Steve, Approximation des équations de Maxwell par des schémas décentrés en éléments finis. Technical Report RR-1601, INRIA (1992).
J.-P. Cioni and M. Remaki, Comparaison de deux méthodes de volumes finis en électromagnétisme. Technical Report RR-3166, INRIA (1997).
J.-P. Cioni, L. Fezoui, L. Anne and F. Poupaud, A parallel FVTD Maxwell solver using 3D unstructured meshes, in 13th annual review of progress in applied computational electromagnetics, Monterey, California (1997).
G. Cohen and P. Joly Eds., Aspects récents en méthodes numériques pour les équations de Maxwell, Collection didactique INRIA, INRIA Rocquencourt, France (1998) 23-27.
S. Depeyre, Étude de schémas d'ordre élevé en volumes finis pour des problèmes hyperboliques. Application aux équations de Maxwell, d'Euler et aux écoulements diphasiques dispersés. Mathématiques appliquées, ENPC, janvier (1997).
R. Eymard, T. Gallouët and R. Herbin, The finite volume method. in Handbook for Numerical Analysis, North Holland-Elsevier Science Publishers, Amsterdam, New York, Oxford (to appear).
Fezoui, L. and Stoufflet, B., A class of implicit upwind schemes for euler simulations with unstructured meshes. J. Comput. Phys. 84 (1989) 174-206. CrossRef
Harten, A., High Resolution Schemes for Hyperbolic Conservation Laws. J. Comput. Phys. 49 (1983) 357-393. CrossRef
Harten, A., Lax, P. D. and van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 36-61. CrossRef
A. Jameson, Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence in transonic and hypersonic flows, in 11th AIAA Computational Fluid Dynamics Conference, Orlando, Florida, July 6-9 (1993), AIAA paper 93-3359.
P. Lesaint, Sur la résolution des systèmes hyperboliques du premier ordre par des méthodes d'éléments finis. Ph.D. thesis, Université de Paris VI, France (1975).
R. Löhner and J. Ambrosiano, A finite element solver for the Maxwell equations, in GAMNI-SMAI Conference on Numerical Methods for the Maxwell Equations, Paris, France (1989). SIAM, Philadelphia (1991).
M. Remaki, A new finite volume scheme for solving Maxwell system. Technical Report RR-3725, INRIA (1999).
M. Remaki, L. Fezoui and F. Poupaud, Un nouveau schéma de type volumes finis appliqué aux équations de Maxwell en milieu hétérogène. Technical Report RR-3351, INRIA (1998).
J.S. Shang, A characteristic-based algorithm for solving 3D, time-domain Maxwell equations. In 30th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 6-9 (1992), AIAA paper 92-0452.
Shang, J.S. and Fithen, R.M., A comparative study of characteristic-based algorithms for the Maxwell equations. J. Comput. Phys. 125 (1996) 378-394. CrossRef
Shankar, V., Hall, W.F. and Mohammadian, A.H., A time-domain differential solver for electromagnetic scattering problems. Proc. IEEE 77 (1989) 709-720. CrossRef
A. Taflove, Re-inventing electromagnetics: supercomputing solution of Maxwell's equations via direct time integration on space grids. AIAA paper 92-0333 (1992).
Umashankar, K.R., Numerical analysis of electromagnetic wave scattering and interaction based on frequency-domain integral equation and method of moments techniques. Wave Motion 10 (1988) 493. CrossRef
Van Leer, B., Towards the ultimate conservative difference scheme v: a second-order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 361-370. CrossRef
Convergence, J.-P. Vila and error estimates in finite volume schemes for general multidimensional scalr conservation laws. I. Explicite monotone schemes. RAIRO Modél. Math. Anal. Numér. 28 (1994) 267-295.
J.-P. Vila and P. Villedieu, Convergence de la méthode des volumes finis pour les systèmes de Friedrichs. C.R. Acad. Sci. Paris Sér. I Math. 3(325) (1997) 671-676.
Warming, R.F. and Hyett, F., The modified equation approach to the stability and accuracy analysis of finite-difference methods. J. Comput. Phys. 14 (1974) 159-179. CrossRef
K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Trans. Antennas and Propagation AP-16 (1966) 302-307.