Abstract
The finite-difference time-domain (FDTD) scheme is one of the most popular computational methods for microwave problems; it is simple to program, highly efficient, and easily adapted to deal with a variety of problems. The FDTD scheme is typically formulated on a structured Cartesian grid and it discretizes Maxwell’s equations formulated in the time domain. The derivatives with respect to space and time are approximated by finite-differences, where the field components of the electric and magnetic field are staggered in space with respect to each other in a particular manner that is tailored for Maxwell’s equations.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
T Abboud, J C Nédélec, and J Volakis. Stable solution of the retarded potential equations. 17th Annual Review of Progress in Applied Computational Electromagnetics, Monterey, CA, pages 146–151, 2001.
M Abramowitz and I A Stegun. Handbook of Mathematical Functions. National Bureau of Standards, 1965.
F Alimenti, P Mezzanotte, L Roselli, and R Sorrentino. A revised formulation of model absorbing and matched modal source boundary conditions for the efficient FDTD analysis of waveguide structures. IEEE Trans. Microwave Theory Tech., 48(1):50–59, January 2000.
O Axelsson. Iterative Solution Methods. New York, NY: Cambridge University Press, 1994.
C A Balanis. Advanced Engineering Electromagnetics. New York, NY: John Wiley & Sons, 1989.
S Balay, W Gropp, L Curfman McInnes, and B Smith. The portable, extensible toolkit for scientific computation. http://www-unix.mcs.anl.gov/petsc/petsc-2/, 2005.
R Barret, M Berry, T F Chan, J Demmel, J Donato, J Dongarra, V Eijkhout, R Pozo, C Romine, and H Van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, PA, 1994. available at: ftp://ftp.netlib.org/templates/templates.ps.
R Beck and R Hiptmair. Multilevel solution of the time-harmonic Maxwell’s equations based on edge elements. Int. J. Numer. Meth. Engng., 45(7):901–920, 1999.
J P Bérenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 114(2):185–200, October 1994.
J Bey. Tetrahedral grid refinement. Computing, 55(4):355–378, 1995.
M J Bluck and S P Walker. Time-domain BIE analysis of large three-dimensional electromagnetic scattering problems. IEEE Trans. Antennas Propagat., 45(5):894–901, May 1997.
Alain Bossavit. Computational Electromagnetism. Boston, MA: Academic Press, 1998.
M M Botha and J M Jin. On the variational formulation of hybrid finite element–boundary integral techniques for electromagnetic analysis. IEEE Trans. Antennas Propagat., 52(11):3037–3047, November 2004.
A C Cangellaris and D B Wright. Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena. IEEE Trans. Antennas Propagat., 39(10):1518–1525, October 1991.
F X Canning. Improved impedance matrix localization method. IEEE Trans. Antennas Propagat., 41(5):659–667, May 1993.
F X Canning and K Rogovin. Fast direct solution of standard moment-method matrices. IEEE Antennas Propagat. Mag., 40(3):15–26, June 1998.
M Celuch-Marcysiak and W K Gwarek. Generalized TLM algorithms with controlled stability margin and their equivalence with finite-difference formulations for modified grids. IEEE Trans. Microwave Theory Tech., 43(9):2081–2089, September 1995.
Z Chen, M M Ney, and W J R Hoefer. A new finite-difference time-domain formulation and its equivalence with the TLM symmetrical condensed node. IEEE Trans. Microwave Theory Tech., 39(12):2160–2169, December 1991.
D K Cheng. Fundamentals of Engineering Electromagnetics. Reading, MA: Addison-Wesley, 1993.
W C Chew, J M Jin, E Michielssen, and J Song. Fast and Efficient Algorithms in Computational Electromagnetics. Norwood, MA: Artech House, 2001.
R Coifman, V Rokhlin, and S Wandzura. The fast multipole method for the wave equation: A pedestrian prescription. IEEE Antennas Propagat. Mag., 35(3):7–12, June 1993.
D B Davidson. Computational Electromagnetics for RF and Microwave Engineering. Cambridge: Cambridge University Press, second edition, 2011.
T Davis. UMFPACK. http://www.cise.ufl.edu/research/sparse/umfpack/, 2005.
J W Demmel, J R Gilbert, and X S Li. SuperLU. http://crd.lbl.gov/~xiaoye/SuperLU/, 2005.
S J Dodson, S P Walker, and M J Bluck. Costs and cost scaling in time-domain integral-equation analysis of electromagnetic scattering. IEEE Antennas Propagat. Mag., 40(4):12–21, August 1998.
R Dyczij-Edlinger and O Biro. A joint vector and scalar potential formulation for driven high frequency problems using hybrid edge and nodal finite elements. IEEE Trans. Microwave Theory Tech., 44(1):15–23, 1996.
R Dyczij-Edlinger, G Peng, and J F Lee. A fast vector-potential method using tangentially continuous vector finite elements. IEEE Trans. Microwave Theory Tech., 46(6):863–868, 1998.
K Eriksson, D Estep, P Hansbo, and C Johnson. Computational Differential Equations. New York, NY: Cambridge University Press, 1996.
R Garg. Analytical and Computational Methods in Electromagnetics. Norwood, MA: Artech House, 2008.
W L Golik. Wavelet packets for fast solution of electromagnetic integral equations. IEEE Trans. Antennas Propagat., 46(5):618–624, May 1998.
R D Graglia. On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle. IEEE Trans. Antennas Propagat., 41(10):1448–1455, October 1993.
D J Griffiths. Introduction to Electrodynamics. Upper Saddle River, NJ: Prentice-Hall, third edition, 1999.
W Hackbusch. Multi-Grid Methods and Application. Berlin: Springer-Verlag, 1985.
W Hackbush. Iterative Solution of Large Sparse Linear Systems of Equations. New York, NY: Springer-Verlag, 1994.
V Hill, O Farle, and R Dyczij-Edlinger. A stabilized multilevel vector finite-element solver for time-harmonic electromagnetic waves. IEEE Trans. Magn., 39(3):1203–1206, 2003.
R Hiptmair. Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal., 36(1):204–225, 1998.
W J R Hoefer. The transmission-line method – theory and applications. IEEE Trans. Microwave Theory Tech., 33(10):882–893, October 1985.
T J R Hughes. The finite element method: linear static and dynamic finite element analysis. Englewood Cliffs, NJ: Prentice-Hall, 1987.
P Ingelström. Higher Order Finite Elements and Adaptivity in Computational Electromagnetics. PhD thesis, Chalmers University of Technology, Göteborg, Sweden, 2004.
J M Jin. The Finite Element Method in Electromagnetics. New York, NY: John Wiley & Sons, 1993.
J M Jin. The Finite Element Method in Electromagnetics. New York, NY: John Wiley & Sons, second edition, 2002.
J M Jin. Theory and Computation of Electromagnetic Fields. New York, NY: John Wiley & Sons, 2010.
P B Johns. A symmetrical condensed node for the TLM method. IEEE Trans. Microwave Theory Tech., 35(4):370–377, April 1987.
G Karypis. METIS. http://www-users.cs.umn.edu/~karypis/metis/, 2005.
P S Kildal, S Rengarajan, and A Moldsvor. Analysis of nearly cylindrical antennas and scattering problems using a spectrum of two-dimensional solutions. IEEE Trans. Antennas Propagat., 44(8):1183–1192, August 1996.
Y Q Liu, A Bondeson, R Bergström, C Johnson, M G Larson, and K Samuelsson. Eddy-current computations using adaptive grids and edge elements. IEEE Trans. Magn., 38(2):449–452, March 2002.
N K Madsen and R W Ziolkowski. A three-dimensional modified finite volume technique for maxwell’s equations. Electromagnetics, 10(1-2):147–161, January-June 1990.
P Monk. Finite Element Methods for Maxwell’s Equations. Oxford: Clarendon Press, 2003.
P B Monk. A comparison of three mixed methods for the time dependent Maxwell equations. SIAM Journal on Scientific and Statistical Computing, 13(5):1097–1122, September 1992.
A Monorchio and R Mittra. A hybrid finite-element finite-difference time-domain (FE/FDTD) technique for solving complex electromagnetic problems. IEEE Microw. Guided Wave Lett., 8(2):93–95, February 1998.
J C Nédélec. Mixed finite elements in R3. Numer. Math., 35(3):315–341, 1980.
N M Newmark. A method of computation for structural dynamics. J. Eng. Mech. Div., Proc. Am. Soc. Civil Eng., 85(EM 3):67–94, July 1959.
S Owen. Meshing Research Corner. http://www.andrew.cmu.edu/user/sowen/mesh.html, 2005.
A F Peterson, S L Ray, and R Mittra. Computational Methods for Electromagnetics. New York, NY: IEEE Press, 1997.
P G Petropoulos, L Zhao, and A C Cangellaris. A reflectionless sponge layer absorbing boundary condition for the solution of Maxwell’s equations with high-order staggered finite difference schemes. J. Comput. Phys., 139(1):184–208, January 1998.
A J Poggio and E K Miller. Integral equation solutions of three-dimensional scattering problems. Computer Techniques for Electromagnetics, Oxford: Pergamon:159–264, 1973.
S M Rao and D R Wilton. Transient scattering by conducting surfaces of arbitrary shape. IEEE Trans. Antennas Propagat., 39(1):56–61, January 1991.
S M Rao, D R Wilton, and A W Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans. Antennas Propagat., AP-30(3):409–418, May 1982.
S Reitzinger and M Kaltenbacher. Algebraic multigrid methods for magnetostatic field problems. IEEE Trans. Magn., 38(2):477–480, 2002.
D J Riley and C D Turner. VOLMAX: A solid-model-based, transient volumetric Maxwell solver using hybrid grids. IEEE Antennas Propagat. Mag., 39(1):20–33, February 1997.
V Rokhlin. Rapid solution of integral equations of classical potential theory. J. Comput. Phys., 60(2):187–207, 1985.
V Rokhlin. Rapid solution of integral equations of scattering theory in two dimensions. J. Comput. Phys., 86(2):414–439, 1990.
T Rylander and A Bondeson. Stability of explicit-implicit hybrid time-stepping schemes for Maxwell’s equations. J. Comput. Phys., 179(2):426–438, July 2002.
T Rylander and J M Jin. Perfectly matched layer for the time domain finite element method. J. of Comput. Phys., 200(1):238–250, October 2004.
T Rylander, T McKelvey, and M Viberg. Estimation of resonant frequencies and quality factors from time domain computations. J. of Comput. Phys., 192(2):523–545, December 2003.
B P Rynne. Instabilities in time marching methods for scattering problems. Electromagnetics, 6(2):129–144, 1986.
Y Saad. Iterative methods for sparse linear systems. Boston, MA: PWS Publishing, 1996.
M N O Sadiku. Numerical Techniques in Electromagnetics with MATLAB. Boca Raton, FL: CRC Press, third edition, 2009.
M Salazar-Palma, T K Sarkar, L E Garcia-Castillo, T Roy, and A Djordjevic. Iterative and Self-Adaptive Finite-Elements in Electromagnetic Modeling. Norwood, MA: Artech House, 1998.
M Schinnerl, J Schöberl, and M Kaltenbacher. Nested multigrid methods for the fast numerical computation of 3D magnetic fields. IEEE Trans. Magn., 36(4):1557–1560, 2000.
R Schuhmann and T Weiland. Stability of the FDTD algorithm on nonorthogonal grids related to the spatial interpolation scheme. IEEE Trans. Magn., 34(5):2751–2754, September 1998.
X Q Sheng and W Song. Essentials of Computational Electromagnetics. Singapore: John Wiley & Sons, 2012.
J R Shewchuk. Trianlge – a two-dimensional quality mesh generator and delaunay triangulator. http://www.cs.cmu.edu/~quake/triangle.html.
J R Shewchuk. Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. Lecture Notes in Computer Science, 1148:203–222, May 1996.
P P Silvester and R L Ferrari. Finite Elements for Electrical Engineers. New York, NY: Cambridge University Press, second edition, 1990.
P D Smith. Instabilities in time marching methods for scattering: cause and rectification. Electromagnetics, 10(4):439–451, October–December 1990.
J M Song and W C Chew. The fast Illinois solver code: requirements and scaling properties. IEEE Comput. Sci. Eng., 5(3):19–23, July–September 1998.
J M Song, C C Lu, and W C Chew. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Trans. Antennas Propagat., 45(10):1488–1493, October 1997.
J M Song, C C Lu, W C Chew, and S W Lee. Fast Illinois solver code (FISC). IEEE Antennas Propagat. Mag., 40(3):27–34, June 1998.
A Taflove. Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 1995.
A Taflove, editor. Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 1998.
A Taflove and S C Hagness. Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, second edition, 2000.
P Thoma and T Weiland. Numerical stability of finite difference time domain methods. IEEE Trans. Magn., 34(5):2740–2743, September 1998.
S Toledo, D Chen, and V Rotkin. TAUCS, A Library of Sparse Linear Solvers. http://www.tau.ac.il/~stoledo/taucs/, 2005.
D A Vechinski and S M Rao. A stable procedure to calculate the transient scattering by conducting surfaces of arbitrary shape. IEEE Trans. Antennas Propagat., 40(6):661–665, June 1992.
R L Wagner and W C Chew. A study of wavelets for the solution of electromagnetic integral equations. IEEE Trans. Antennas Propagat., 43(8):802–810, August 1995.
J J H Wang. Generalized Moment Methods in Electromagnetics. New York, NY: John Wiley & Sons, 1991.
K F Warnick. NUMERICAL METHODS FOR ENGINEERING - An Introduction Using MATLAB and Computational Electromagnetics Examples. Raleigh, NC: SciTech Publishing, 2011.
J P Webb. Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements. IEEE. Trans. Antennas Propagat., 47(8):1244–1253, 1999.
T Weiland. Time domain electromagnetic field computation with finite difference methods. Int. J. Numer. Model. El., 9(4):295–319, July-August 1996.
P Wesseling. An Introduction to Multigrid Methods. Chichester: John Wiley & Sons, 1992.
R B Wu and T Itoh. Hybrid finite-difference time-domain modeling of curved surfaces using tetrahedral edge elements. IEEE Trans. Antennas Propagat., 45(8):1302–1309, August 1997.
K S Yee. Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propagat., AP-14(3):302–307, May 1966.
K S Yee and J S Chen. The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving Maxwell’s equations. IEEE Trans. Antennas Propagat., 45(3):354–363, March 1997.
K S Yee, J S Chen, and A H Chang. Numerical experiments on PEC boundary condition and late time growth involving the FDTD/FDTD and FDTD/FVTD hybrid. IEEE Antennas Propagat. Soc. Int. Symp., 1:624–627, 1995.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Rylander, T., Ingelström, P., Bondeson, A. (2013). The Finite-Difference Time-Domain Method. In: Computational Electromagnetics. Texts in Applied Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5351-2_5
Download citation
DOI: https://doi.org/10.1007/978-1-4614-5351-2_5
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-5350-5
Online ISBN: 978-1-4614-5351-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)