×

Conservative space-time mesh refinement methods for the FDTD solution of Maxwell’s equations. (English) Zbl 1107.78015

The paper focuses on a special time discretization approach formulated by the authors in order to keep subgridding schemes in Finite Difference Time Domain procedures conservative. They claim very general applicability though they demonstrate their method only for the Maxwellian wave propagation problem with a spatial 1-2 refinement within a rectangular subdomain of the discretized volume. Time stepping within the subdomain is refined by a factor of two as well. Coupling between subgrided and exterior domain is provided by the mean of a virtual current density discretized on the separating surface, which is calculated every (exterior) time step by inverting a sparse linear system.
At least at this point remarks about the nature of alternative coupling quantities for other kind of problems would be helpful with respect to the generality claimed in the paper. Proofs of existence of the discretized solution, energy conservation and stability – under given conditions – are described in detail in the paper. The authors present numerical experiments that are appropriate to demonstrate the function of the procedure, whereas some conclusions according comparisons between results from overall refinements and the approach under discussion could have been drawn in a more differentiated manner. At least some work of the group around T. Weiland, e.g. of P. Thoma and T. Weiland, is missing in the reference list.

MSC:

78M20 Finite difference methods applied to problems in optics and electromagnetic theory
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
78A40 Waves and radiation in optics and electromagnetic theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bamberger, A.; Glowinski, R.; Tran, Q. H., A domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change, SIAM J. Numer. Anal., 34, 2, 603-639 (1997) · Zbl 0877.35066
[2] Bécache, E.; Joly, P.; Rodríguez, J., Space-time mesh refinement for elastodynamics. Numerical results, Comput. Meth. Appl. Mech. Eng., 194, 2-5, 355-366 (2005) · Zbl 1095.74030
[3] Belgacem, F. B.; Buffa, A.; Maday, Y., The mortar finite element method for 3d Maxwell equations: first results, SIAM J. Numer. Anal., 39, 3, 880-901 (2001) · Zbl 1001.65123
[4] Belgacem, F. B., The mortar finite element method with Lagrange multipliers, Numer. Math., 84, 173-197 (1999) · Zbl 0944.65114
[5] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods. Mixed and hybrid finite element methods, Springer series in computational mathematics, vol. 15 (1991), Springer: Springer Berlin · Zbl 0788.73002
[6] Buffa, A.; Ciarlet, P., On traces for functional spaces related to Maxwell’s equations. ii. Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Meth. Appl. Sci., 24, 1, 31-48 (2001) · Zbl 0976.46023
[7] Chevalier, M. W.; Luebbers, R. J., FDTD local grid with material traverse, IEEE Trans. Antenn. Propagat., 45, 3, 411-421 (1997)
[8] Cohen, G., Higher-order numerical methods for transient wave equations. Scientific computation (2002), Springer: Springer Berlin, New York
[9] F. Collino, T. Fouquet, P. Joly. Analyse numérique d’une méthode de raffinement de maillage espace-temps pour l’équation des ondes, Technical Report 3474, INRIA, Aout, 1998.; F. Collino, T. Fouquet, P. Joly. Analyse numérique d’une méthode de raffinement de maillage espace-temps pour l’équation des ondes, Technical Report 3474, INRIA, Aout, 1998.
[10] Collino, F.; Fouquet, T.; Joly, P., Une méthode de raffinement de maillage espace-temps pour le système de maxwell en dimension un, CR Acad. Sci. Paris, 328, 263-268 (1999) · Zbl 0934.78002
[11] Collino, F.; Fouquet, T.; Joly, P., A conservative space-time mesh refinement method for the 1-d wave equation. Part I: Construction, Numerische mathematik, 95, 197-221 (2003) · Zbl 1048.65089
[12] Collino, F.; Fouquet, T.; Joly, P., A conservative space-time mesh refinement method for the 1-d wave equation. Part II: Analysis, Numerische mathematik, 95, 223-251 (2003) · Zbl 1036.65073
[13] Collino, F.; Joly, P.; Millot, F., Fictitious domain method for unsteady problems: application to electromagnetic scattering, J. Comput. Phys., 138, 907-938 (1997) · Zbl 1126.78311
[14] T. Fouquet, Raffinement de maillage spatio temporel pour les équations de Maxwell, Ph.D. Thesis, Université Paris IX Dauphine, June, 2000.; T. Fouquet, Raffinement de maillage spatio temporel pour les équations de Maxwell, Ph.D. Thesis, Université Paris IX Dauphine, June, 2000.
[15] Gedney, S. D.; Lansing, F., Computational electrodynamics the finite-difference time-domain method, (Taflove, A., Chapter explicit time-domain solution of Maxwell’s equations using nonorthogonal and unstructured grids (1995), Artech House: Artech House Boston, London), 343-393
[16] Joly, P., Variational methods for time-dependent wave propagation problems, (Computational wave propagation. Direct and inverse problems (2003), Springer: Springer Berlin), 201-264 · Zbl 1049.78028
[17] P. Joly, J. Rodríguez, An error analysis of conservative space-time mesh refinement methods for the 1-d wave equation (submitted).; P. Joly, J. Rodríguez, An error analysis of conservative space-time mesh refinement methods for the 1-d wave equation (submitted).
[18] Kim, I. S.; Hoefer, W. J.R., A local mesh refinement algorithm for the time-domain finite-difference method to solve Maxwell’s equations, IEEE Trans. Microwave Theory Tech., 38, 6, 812-815 (1990)
[19] Kunz, K. S.; Simpson, L., A technique for increasing the resolution of finite-difference solutions to the Maxwell equations, IEEE Trans. Electromagn. Compat., EMC-23, 419-422 (1981)
[20] Monk, P., Sub-gridding FDTD schemes, ACES J., 11, 37-46 (1996)
[21] Nedelec, J., A new family of mixed finite elements in \(R^3\), Numer. Math., 50, 57-81 (1986) · Zbl 0625.65107
[22] Prescott, D. T.; Shuley, N. V., A method for incorporating different sized cells into the finite-difference time-domain analysis technique, IEEE Microwave Guided Wave Lett., 2, 434-436 (1992)
[23] Yee, K., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antenn. Propagat., 302-307 (1966) · Zbl 1155.78304
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.