×

Arbitrary high order finite-volume methods for electromagnetic wave propagation. (English) Zbl 1196.65145

Summary: Problems in electromagnetic wave propagation often require high accuracy approximations with low resolution computational grids. For non-stationary problems such schemes should possess the same approximation order in space and time. In the present article we propose for electromagnetic applications an explicit class of robust finite-volume (FV) schemes for the Maxwell equations. To achieve high accuracy we combine the FV method with the so-called ADER approach resulting in schemes which are arbitrary high order accurate in space and time. Numerical results and convergence investigations are shown for two and three-dimensional test cases on Cartesian grids, where the used FV-ADER schemes are up to 8th order accurate in both space and time.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
78M12 Finite volume methods, finite integration techniques applied to problems in optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abn El-Raouf, H. E.; El-Diwani, E. A.; Abn El-Hadi, A.; El-Hefnawi, F. M., A low-dispersion 3-D second-order in time fourth-order in space FDTD scheme (M3d24), IEEE Trans. Antennas Propagation, 52, 1638-1646 (2004) · Zbl 1368.78165
[2] Butcher, J., The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (1987), Wiley: Wiley New York · Zbl 0616.65072
[3] Cole, J., High-accuracy Yee algorithm based on nonstandard finite differences: New developments and verifications, IEEE Trans. Antennas Propagation, 50, 1185-1191 (2002) · Zbl 1368.65134
[4] M. Dumbser and C.-D. Munz, Building blocks for arbitrary high order discontinuous Galerkin schemes, J. Sci. Comput. (2005), in press; M. Dumbser and C.-D. Munz, Building blocks for arbitrary high order discontinuous Galerkin schemes, J. Sci. Comput. (2005), in press · Zbl 1210.65165
[5] Givoli, D.; Neta, B., High-order non-reflecting boundary scheme for time-dependent waves, J. Comput. Phys., 186, 24-46 (2003) · Zbl 1025.65049
[6] Hu, F. Q., A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables, J. Comput. Phys., 173, 455-480 (2001) · Zbl 1051.76593
[7] Jackson, J. D., Classical Electrodynamics (1975), Wiley: Wiley New York · Zbl 0114.42903
[8] Kröner, D., Numerical Schemes for Conservation Laws (1997), Wiley Teubner: Wiley Teubner Chichester-Leipzig · Zbl 0872.76001
[9] LeVeque, R., Numerical Methods for Conservation Laws (1990), Birkhäuser: Birkhäuser Basel · Zbl 0723.65067
[10] Madsen, N. K.; Ziolkowski, R. W., A three-dimensional modified finite volume technique for Maxwell’s equations, Electromagnetics, 10, 147-161 (1990)
[11] Marchuk, G., Methods of Numerical Mathematics (1975), Springer-Verlag: Springer-Verlag New York · Zbl 0329.65002
[12] Millington, R.; Toro, E.; Nejad, L., Arbitrary High Order Methods For Conservation Laws I: The One Dimensional Case (1999), Department of Computing and Mathematics: Department of Computing and Mathematics Manchester Metropolitan University, Chester Street, Manchester, M1 5GD, UK
[13] Mohammadian, V. S.A. H.; Hall, W., Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure, Comput. Phys. Comm., 68, 175-196 (1991)
[14] Munz, C.-D.; Omnes, P.; Schneider, R., A three-dimensional finite-volume solver for Maxwell equations with divergence cleaning on unstructured meshes, Comput. Phys. Comm., 130, 83-117 (2000) · Zbl 0960.78019
[15] Munz, C.-D.; Schneider, R.; Voß, U., A finite-volume method for Maxwell equations in time domain, SIAM J. Sci. Comput., 22, 449-475 (2000) · Zbl 1039.78012
[16] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T., Numerical Recipes (1987), Cambridge University Press: Cambridge University Press Cambridge
[17] Roe, P., Modern shock-capturing schemes, (Takayama, K., Proc. of 18th Internat. Symp. on Shock Waves (1991), Springer-Verlag: Springer-Verlag Berlin), 29-40
[18] Schwartzkopff, T.; Dumbser, M.; Munz, C.-D., Fast high order ADER schemes for linear hyperbolic equations, J. Comput. Phys., 197, 532-539 (2004) · Zbl 1052.65078
[19] Schwartzkopff, T.; Dumbser, M.; Munz, C.-D., Arbitrary high order finite volume schemes for linear wave propagation, (Hirschel, E. H.; etal., Notes on Numerical Fluid Mechanics and Multidisciplinary Design (2005), Springer: Springer Berlin) · Zbl 1196.65145
[20] T. Schwartzkopff, C.-D. Munz, E. Toro, R. Millington, ADER-2d: A very high-order approach for linear hyperbolic systems, in: Proc. of ECCOMAS CFD, Swansea, UK, ISBN 0 905 091 12 4, 2001; T. Schwartzkopff, C.-D. Munz, E. Toro, R. Millington, ADER-2d: A very high-order approach for linear hyperbolic systems, in: Proc. of ECCOMAS CFD, Swansea, UK, ISBN 0 905 091 12 4, 2001 · Zbl 1028.76030
[21] Schwartzkopff, T.; Munz, C.-D.; Toro, E.; Millington, R., ADER: A high order approach for linear hyperbolic systems in 2d, J. Sci. Comput., 17, 231-240 (2002) · Zbl 1022.76034
[22] C.-W. Shu, Essentially non-oscillatory and weighted non-oscillatory schemes for hyperbolic conservation laws, NASA/CR-97-206253 ICASE Report No. 97-65, 1997; C.-W. Shu, Essentially non-oscillatory and weighted non-oscillatory schemes for hyperbolic conservation laws, NASA/CR-97-206253 ICASE Report No. 97-65, 1997
[23] Sonar, T., Methods on unstructured grids, WENO and ENO recovery techniques, (Meister, A.; Struckmeier, J., Hyperbolic Partial Differential Equations (2002), Vieweg-Verlag), 115-232
[24] Taflove, A., Advances in Computational Electrodynamics (1998), Artech House: Artech House Boston · Zbl 0903.65098
[25] Toro, E., Riemann Solvers and Numerical Methods for Fluid Dynamics (1997), Springer: Springer Berlin
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.