A high-order accurate parallel solver for Maxwell's equations on overlapping grids

SIAM Journal on Scientific Computing

Volumn 28, Issue 5, 2006, Pages 1730-1765

  Henshaw, William D ^a  

^a LAWRENCE LIVERMORE NATIONAL LABORATORY   (United States)

Author keywords

High order accurate; Maxwell's equations; Overlapping grids; Symmetric finite difference

Indexed keywords

BOUNDARY CONDITIONS; GRID COMPUTING; MATHEMATICAL OPERATORS; MAXWELL EQUATIONS; WAVE EQUATIONS;

DISCRETE BOUNDARY CONDITIONS; LAPLACE OPERATOR; OVERLAPPING GRIDS; TIME-STEPPING RESTRICTION;

PROBLEM SOLVING;

EID: 34648836550     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/050644379     Document Type: Article

References (45)

1. L. ANNÉ, P. JOLY, AND Q. H. TRAN, Construction and analysis of higher order finite difference schemes for the ID wave equation, Comput. Geosci., 4 (2000), pp. 207-249.
2. C. A. BALANIS, Advanced Engineering Electromagnetics, John Wiley & Sons, New York, 1989.
3. G. CHESSHIRE AND W. HENSHAW, Composite overlapping meshes for the solution of partial differential equations, J. Comput. Phys., 90 (1990), pp. 1-64.
4. G. C. COHEN, Higher-Order Numerical Methods for Transient Wave Equations, Springer, New York, 2002.
5. M. COSTABLE AND M. DAUGE, Weighted regularization of Maxwell equations in polyhedral domains, Numer. Math., 93 (2002), pp. 239-277.
6. M. DABLAIN, The application of high order differencing for the scalar wave equation, Geophysics, 51 (1986), pp. 54-66.
7. S. DEY AND R. MITTRA, A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects, IEEE Microwave Guided Wave Lett., 7 (1997), pp. 273-275.
8. A.-S. B.-B. DHIA, C. HAZARD, AND S. LOHRENGEL, A singular field method for the solution of Maxwell's equations in polyhedral domains, SIAM J. Appl. Math., 59 (1999), pp. 2028-2044.
9. A. DITKOWKSI, K. DRIDI, AND J. HESTHAVEN, Convergent Cartesian grid methods for Maxwell's equations in complex geometries, J. Comput. Phys., 170 (2001), pp. 39-80.
10. T. A. DRISCOLL AND B. FORNBERG, A block pseudospectral method for Maxwell's equations. I. One-dimensional case, J. Comput. Phys., 140 (1998), pp. 47-65.
11. T. A. DRISCOLL AND B. FORNBERG, Block pseudospectral methods for Maxwell's equations II: Two-dimensional, discontinuous-coefficient case, SIAM J. Sci. Comput., 21 (1999), pp. 1146-1167.
12. B. FORNBERG, A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge, UK, 1996.
13. S. D. GEDNEY, F. S. LANSING, AND D. L. RASCOE, Full wave analysis of microwave monolithic circuit devices using a generalized Yee-algorithm based on an unstructured grid, IEEE Trans. Microwave Theory Tech., 44 (1996), pp. 1393-1400.
14. B. GUSTAFSSON, H.-O. KREISS, AND J. OLIGER, Time Dependent Problems and Difference Methods, John Wiley & Sons, New York, 1995.
15. B. GUSTAFSSON AND E. MOSSBERG, Time compact high order difference methods for wave propagation, SIAM J. Sci. Comput., 26 (2004), pp. 259-271.
16. B. GUSTAFSSON AND P. WALHUND, Time compact difference methods for wave propagation in discontinuous media, SIAM J. Sci. Comput., 26 (2004), pp. 272-293.
17. C. HAZARD, Numerical simulation of corner singularities: A paradox in Maxwell-like problems, C. R. Mecanique, 330 (2002), pp. 57-68.
18. C. HAZARD AND S. LOHRENGEL, A singular field method for Maxwell's equations: Numerical aspects for 2D magnetostatics, SIAM J. Numer. Anal., 40 (2002), pp. 1021-1040.
19. W. HENSHAW, Mappings for Overture, a Description of the Mapping Class and Documentation for Many Useful Mappings, Research report UCRL-MA-132239, Lawrence Livermore National Laboratory, Livermore, CA, 1998.
20. W. HENSHAW, H.-O. KREISS, AND L. REYNA, A fourth-order accurate difference approximation for the incompressible Navier-Stokes equations, Comput. & Fluids, 23 (1994), pp. 575-593.
21. W. D. HENSHAW, On multigrid for overlapping grids, SIAM J. Sci. Comput., 26 (2005), pp. 1547-1572.
22. W. D. HENSHAW AND D. W. SCHWENDEMAN, An adaptive numerical scheme for high-speed reactive flow on overlapping grids, J. Comput. Phys., 191 (2003), pp. 420-447.
23. J. HESTHAVEN AND T. WARBURTON, Nodal high-order methods on unstructured grids, I. Time-domain solution of Maxwell's equations, J. Comput. Phys., 181 (2002), pp. 186-221.
24. R. HOLLAND, Finite difference solution of Maxwell's equations in generalized nonorthogonal coordinates, IEEE Trans. Nuclear Sci., 30 (1983), pp. 4589-4591.
25. J. JIN, The Finite Element Method in Electromagnetics, John Wiley & Sons, New York, 1993.
26. H. M. JURGENS AND D. W. ZINGG, Numerical solution of the time-domain Maxwell equations using high-accuracy finite-difference methods, SIAM J. Sci. Comput., 22 (2000), pp. 1675-1696.
27. H.-O. KREISS AND N. A. PETERSSON, A second order accurate embedded boundary method for the wave equation with Dirichlet data, SIAM J. Sci. Comput., 27 (2006), pp. 1141-1167.
28. H.-O. KREISS, N. A. PETERSSON, AND J. YSTRÖM, Difference approximations for the second order wave equation, SIAM J. Numer. Anal., 40 (2002), pp. 1940-1967.
29. J. LAMBERT, Computational Methods in Ordinary Differential Equations, Cambridge University Press, New York, 1973.
30. J. LEE, R. PALENDECH, AND R. MITTRA, Modeling three-dimensional discontinuities in wave-guides using nonorthogonal FDTD algorithm, IEEE Trans. Microwave Theory Tech., 40 (1992), pp. 346-352.
31. R. B. LEHOUCQ, D. C. SORENSEN, AND C. YANG, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia, 1998.
32. N. K. MADSEN, Divergence preserving discrete surface integral method for Maxwell's curl equations using non-orthogonal unstructured grids, J. Comput. Phys., 119 (1995), pp. 34-45.
33. 3, Numer. Math., 35 (1980), pp. 315-341.
34. D. QUINLAN, A++/P++ Class Libraries, Research report LA-UR-95-3273, Los Alamos National Laboratory, Los Alamos, NM, 1995.
35. L. G. M. REYNA, Part III. On Composite Meshes, Ph.D. thesis, Department of Applied Mathematics, California Institute of Technology, Pasadena, CA, 1982.
36. P. ROACHE, Code verification by the method of manufactured solutions, ASME J. Fluids Eng., 124 (2002), pp. 4-10.
37. G. RODRIGUE AND D. WHITE, A vector finite element time-domain method for solving Maxwell's equations on unstructured hexahedral grids, SIAM J. Sci. Comput., 23 (2001), pp. 683-706.
38. T. RYLANDER AND A. BONDESON, Application of stable FEM-FDTD hybrid to scattering problems, IEEE Trans. Antennas and Propagation, 50 (2002), pp. 141-144.
39. G. R. SHUBIN AND J. B. BELL, A modified equation approach to constructing fourth order methods for acoustic wave propagation, SIAM J. Sci. Statist. Comput., 8 (1987), pp. 135-151.
40. A. SUSSMAN, G. AGRAWAL, AND J. SALTZ, A Manual for the Multiblock PARTI Runtime Primitives, Revision 4.1, Technical report CS-TR-3070.1, Department of Computer Science, University of Maryland, College Park, MD, 1993.
41. A. TAFLOVE AND S. C. HAGNESS, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Boston, 2000.
42. H. VAN DE HULST, Light Scattering by Small Particles, Dover, New York, 1957.
43. B. YANG, D. GOTTLIEB, AND J. HESTHAVEN, Spectral simulations of electromagnetic wave scattering, J. Comput. Phys., 134 (1997), pp. 216-230.
44. K. YEE, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Trans. Antennas and Propagation, 14 (1966), pp. 302-307.
45. K. S. YEE, J. S. CHEN, AND A. H. CHANG, Conformal finite-difference time-domain (FDTD) with overlapping grids, IEEE Trans. Antennas and Propagation, 40 (1992), pp. 1068-1075.