An explicit fourth-order orthogonal curvilinear staggered-grid FDTD method for Maxwell's equations

The explicit fourth-order staggered finite-difference time-domain scheme, previously proposed for a Cartesian grid, is extended to Maxwell's equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems. A simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem. Numerical experiments are presented to illustrate the efficiency and accuracy of the method.

[1]  A. Allievi,et al.  Application of Bubnov-Galerkin formulation to orthogonal grid generation , 1992 .

[2]  S.,et al.  Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media , 1966 .

[3]  Joe F. Thompson,et al.  Numerical grid generation: Foundations and applications , 1985 .

[4]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[5]  Eli Turkel,et al.  A fourth-order accurate finite-difference scheme for the computation of elastic waves , 1986 .

[6]  R. Löhner,et al.  Electromagnetics via the Taylor-Galerkin Finite Element Method on Unstructured Grids , 1994 .

[7]  D. Gottlieb,et al.  Stable and accurate boundary treatments for compact, high-order finite-difference schemes , 1993 .

[8]  R. Holland.  Finite-Difference Solution of Maxwell's Equations in Generalized Nonorthogonal Coordinates , 1983, IEEE Transactions on Nuclear Science.

[9]  Tobin A. Driscoll,et al.  Block Pseudospectral Methods for Maxwell's Equations II: Two-Dimensional, Discontinuous-Coefficient Case , 1999, SIAM J. Sci. Comput..

[10]  J. S. Chen,et al.  The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving Maxwell's equations , 1997 .

[11]  F. W. Kellaway,et al.  Advanced Engineering Mathematics , 1969, The Mathematical Gazette.

[12]  Leung Tsang,et al.  Electromagnetic scattering of waves by random rough surface: A finite-difference time-domain approach , 1991 .

[13]  Richard W. Ziolkowski,et al.  Numerical solution of Maxwell's equations in the time domain using irregular nonorthogonal grids , 1988 .

[14]  Jeffrey L. Young,et al.  Toward the construction of a fourth-order difference scheme for transient EM wave simulation: staggered grid approach , 1997 .

[15]  Willy Hereman,et al.  Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces , 1998 .

[16]  Jiayuan Fang,et al.  A locally conformed finite-difference time-domain algorithm of modeling arbitrary shape planar metal strips , 1993 .

[17]  Hiroyuki Ichikawa,et al.  Electromagnetic analysis of diffraction gratings by the finite-difference time-domain method , 1998 .

[18]  Allen Taflove,et al.  Application of the Finite-Difference Time-Domain Method to Sinusoidal Steady-State Electromagnetic-Penetration Problems , 1980, IEEE Transactions on Electromagnetic Compatibility.

[19]  Patrick M. Knupp,et al.  Fundamentals of Grid Generation , 2020 .

[20]  A Laplacian equation method for numerical generation of boundary-fitted 3D orthogonal grids , 1989 .

[21]  Bertil Gustafsson,et al.  The convergence rate for difference approximations to general mixed initial boundary value problems , 1981 .

[22]  Yanfen Hao,et al.  Analyzing electromagnetic structures with curved boundaries on Cartesian FDTD meshes , 1998 .

[23]  J. Plumey,et al.  Generalization of the coordinate transformation method with application to surface-relief gratings , 1999 .

[24]  A. Cangellaris,et al.  Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena , 1991 .

[25]  R. J. Joseph,et al.  Advances in Computational Electrodynamics: The Finite - Di erence Time - Domain Method , 1998 .

[26]  A. Yefet,et al.  A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell''s equations , 1999 .

[27]  R. LeVeque,et al.  The immersed interface method for acoustic wave equations with discontinuous coefficients , 1997 .

[28]  R. Holland,et al.  Pitfalls of staircase meshing , 1993 .

[29]  Joe F. Thompson,et al.  Numerical grid generation , 1985 .

[30]  Eli Turkel,et al.  Fourth order compact implicit method for the Maxwell equations with discontinuous coefficients , 2000 .

[31]  Jin Au Kong,et al.  Wave scattering from a periodic dielectric surface for a general angle of incidence , 1982 .

[32]  M. Fusco,et al.  FDTD algorithm in curvilinear coordinates (EM scattering) , 1990 .

[33]  K. S. Yee,et al.  Conformal finite-different time-domain (FDTD) with overlapping grids , 1992 .

[34]  D. Gottlieb,et al.  Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes , 1994 .

[35]  D. Pathria,et al.  The Correct Formulation of Intermediate Boundary Conditions for Runge-Kutta Time Integration of Initial Boundary Value Problems , 1997, SIAM J. Sci. Comput..

[36]  H. Kreiss,et al.  Numerical solution of the coupled mode equations in duct acoustics , 1994 .

[37]  Leif Abrahamsson,et al.  Orthogonal grid generation for two-dimensional ducts , 1991 .

[38]  Eli Turkel,et al.  On the construction of a high order difference scheme for complex domains in a Cartesian grid , 2000 .

[39]  Andreas C. Cangellaris,et al.  A Reflectionless Sponge Layer Absorbing Boundary Condition for the Solution of Maxwell's Equations with High-Order Staggered Finite Difference Schemes , 1998 .

[40]  David Gottlieb,et al.  On the construction and analysis of absorbing layers in CEM , 1998 .

[41]  K. S. Yee,et al.  Conformal finite difference time domain (FDTD) with overlapping grids , 1992, IEEE Antennas and Propagation Society International Symposium 1992 Digest.

[42]  Dennis N. Assanis,et al.  Generation of orthogonal grids with control of spacing , 1991 .

[43]  Elisabeth Larsson,et al.  Iterative Solution of the Helmholtz Equation by a Second-Order Method , 1999, SIAM J. Matrix Anal. Appl..

[44]  B. Gustafsson.  The convergence rate for difference approximations to mixed initial boundary value problems , 1975 .

[45]  Andrea Prosperetti,et al.  Orthogonal mapping in two dimensions , 1992 .

[46]  Bo Zhang,et al.  An explicit fourth-order staggered finite-difference time-domain method for Maxwell's equations , 2002 .

[47]  A. Taflove,et al.  Numerical Solution of Steady-State Electromagnetic Scattering Problems Using the Time-Dependent Maxwell's Equations , 1975 .

[48]  A noniterative method for the generation of orthogonal coordinates in doubly-connected regions , 1982 .

[49]  D. Maystre,et al.  Waterman and Rayleigh methods for diffraction grating problems: extension of the convergence domain , 1998 .

[50]  Allen Taflove,et al.  Computational Electrodynamics the Finite-Difference Time-Domain Method , 1995 .

[51]  Jin Au Kong,et al.  A Finite-Difference Time-Domain Analysis of Wave Scattering from Periodic Surfaces: Oblique Incidence Case , 1993 .

[52]  Erwin Kreyszig,et al.  Advanced Engineering Mathematics, Maple Computer Guide , 2000 .

[53]  Patrick Joly,et al.  Construction and Analysis of Fourth-Order Finite Difference Schemes for the Acoustic Wave Equation in Nonhomogeneous Media , 1996 .

[54]  N. Madsen.  Divergence preserving discrete surface integral methods for Maxwell's curl equations using non-orthogonal unstructured grids , 1995 .

[55]  David I. Gottlieb,et al.  The Theoretical Accuracy of Runge-Kutta Time Discretizations for the Initial Boundary Value Problem: A Study of the Boundary Error , 1995, SIAM J. Sci. Comput..

[56]  Allen Taflove,et al.  Finite-difference time-domain modeling of curved surfaces (EM scattering) , 1992 .

[57]  Amir Yefet,et al.  A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations , 2001 .

[58]  Raj Mittra,et al.  Modeling three-dimensional discontinuities in waveguides using nonorthogonal FDTD algorithm , 1992 .

[59]  Elisabeth Larsson,et al.  A Domain Decomposition Method for the Helmholtz Equation in a Multilayer Domain , 1999, SIAM J. Sci. Comput..

[60]  K. Yee.  Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media , 1966 .

[61]  John B. Schneider,et al.  A Monte-Carlo FDTD technique for rough surface scattering , 1995 .