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ADI-FDTD Method for Two-Dimensional Transient Electromagnetic Problems

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  15 January 2016

Wanshan Li ,
Yile Zhang ,
Yau Shu Wong  and
Dong Liang
Wanshan Li
Affiliation:
School of Mathematics, Shandong University, Jinan 250199, P.R. China
Yile Zhang*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada
Yau Shu Wong
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada
Dong Liang
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada
*
*Corresponding author. Email addresses:[email protected] (W. Li), [email protected] (Y. Zhang), [email protected] (Y. S. Wong), [email protected] (D. Liang)
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Abstract

An efficient and accurate numerical scheme is proposed for solving the transverse electric (TE) mode electromagnetic (EM) propagation problem in two-dimensional earth. The scheme is based on the alternating direction finite-difference time-domain (ADI-FDTD) method. Unlike the conventional upward continuation approach for the earth-air interface, an integral formulation for the interface boundary is developed and it is effectively incorporated to the ADI solver. Stability and convergence analysis together with an error estimate are presented. Numerical simulations are carried out to validate the proposed method, and the advantage of the present method over the popular Du-Fort-Frankel scheme is clearly demonstrated. Examples of the electromagnetic field propagation in the ground with anomaly further verify the effectiveness of the proposed scheme.

Keywords

ADI-FDTDinterface boundarystability and convergence analysis

MSC classification

Secondary: 65N06: Finite difference methods 65N12: Stability and convergence of numerical methods 65N15: Error bounds 65N22: Solution of discretized equations
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 94 - 123
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Adhidjaja, J. I., Hohmann, G. W. and Oristaglio, M. L., Two-dimensional transient electromagnetic responses, Geophysics, 50 (1985), 2849–2861.Google Scholar
[2]Chen, W., Li, X. and Liang, D., Symmetric energy-conserved splitting FDTD scheme for the Maxwell's equations, Comm. Comput. Phys., 6 (2009), 804–825.Google Scholar
[3]Chung, Y., Son, J.-S., Lee, T. J., Kim, H. J. and Shin, C., Three-dimensional modelling of controlled-source electromagnetic surveys using an edge finite-element method with a direct solver, Geophys. Prospect., 62 (2014), 1468–1483.Google Scholar
[4]Commer, M. and Newman, G., A parallel finite-difference approach for 3D transient electromagnetic modeling with galvanic sources, Geophysics, 69 (2004), 1192–1202.Google Scholar
[5]Constable, S. and Weiss, C. J., Mapping thin resistors and hydrocarbons with marine EM methods: Insights from 1D modeling, Geophysics, 71 (2006), G43–G51.Google Scholar
[6]da Silva, N. V., Morgan, J. V., MacGregor, L. and Warner, M., A finite element multifrontal method for 3D CSEM modeling in the frequency domain, Geophysics, 77 (2012), E101–E115.Google Scholar
[7]Everett, M. and Edwards, R., Transient marine electromagnetics: the 2.5-D forward problem, Geophys. J. Int., 113 (1993), 545–561.Google Scholar
[8]Freund, R. W. and Nachtigal, N. M., An implementation of the QMR method based on coupled two-term recurrences, SIAM J. Sci. Comput., 15 (1994), 313–337.Google Scholar
[9]Gao, L. and Liang, D., New energy-conserved identities and super-convergence of the symmetric EC-S-FDTD scheme for Maxwell's equations in 2D, Comm. Comput. Phys., 11 (2012), 1673–1696.Google Scholar
[10]Goldman, Y., Hubans, C., Nicoletis, S. and Spitz, S., A finite-element solution for the transient electromagnetic response of an arbitrary two-dimensional resistivity distribution, Geophysics, 51 (1986), 1450–1461.Google Scholar
[11]Grayver, A., Streich, R. and Ritter, O., Three-dimensional parallel distributed inversion of CSEM data using a direct forward solver, Geophys. J. Int., 193 (2013), 1432–1446.Google Scholar
[12]Grayver, A. V. and Bürg, M., Robust and scalable 3-D geo-electromagnetic modelling approach using the finite element method, Geophys.J. Int., 198 (2014), 110–125.Google Scholar
[13]Gronwall, T. H., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 20 (1919), 292–296.Google Scholar
[14]Haber, E. and Ascher, U. M., Fast finite volume simulation of 3D electromagnetic problems with highly discontinuous coefficients, SIAM J. Sci. Comput., 22 (2001), 1943–1961.Google Scholar
[15]Haber, E. and Schwarzbach, C., Parallel inversion of large-scale airborne time-domain electromagnetic data with multiple octree meshes, Inverse Prob., 30 (2014), 055011.Google Scholar
[16]Huang, J. and Wood, A., Analysis and numerical solution of transient electromagnetic scattering from overfilled cavities, Comm. Comput. Phys., 1 (2006), 1043–1055.Google Scholar
[17]Huang, J., Wood, A. W. and Havrilla, M.J., A hybrid finite element-Laplace transform method for the analysis of transient electromagnetic scattering by an over-filled cavity in the ground plane, Comm. Comput. Phys., 5 (2009), 126–141.Google Scholar
[18]Jiang, X., Zhang, L. and Zheng, W., Adaptive hp-finite element computations for time-harmonic Maxwells equations, Comm. Comput. Phys., 13 (2013), 559–582.Google Scholar
[19]Jim, Douglas Jr., On the numerical integration by implicit methods, J. Soc. Ind. Appl. Math., 3 (1955), 42–65.Google Scholar
[20]Key, K. and Ovall, J., A parallel goal-oriented adaptive finite element method for 2.5-D electromagnetic modelling, Geophys. J. Int., 186 (2011), 137–154.Google Scholar
[21]Koldan, J., Puzyrev, V., de la Puente, J., Houzeaux, G. and Cela, J. M., Algebraic multigrid pre-conditioning within parallel finite-element solvers for 3-D electromagnetic modelling problems in geophysics, Geophys. J. Int., 197 (2014), 1442–1458.Google Scholar
[22]Kuo, J. T. and Cho, D.-h., Transient time-domain electromagnetics, Geophysics, 45 (1980), 271–291.Google Scholar
[23]Lee, K. H., Electromagnetic scattering by two-dimensional inhomogeneity due to an oscillating magnetic dipole, California Univ., Berkeley (USA), 1978.Google Scholar
[24]Leppin, M., Electromagnetic modeling of 3-D sources over 2-D inhomogeneities in the time domain, Geophysics, 57 (1992), 994–1003.Google Scholar
[25]Maaø, R. A., Fast finite-difference time-domain modeling for marine-subsurface electromagnetic problems, Geophysics, 72 (2007), A19–A23.Google Scholar
[26]Mukherjee, S. and Everett, M. E., 3D controlled-source electromagnetic edge-based finite element modeling of conductive and permeable heterogeneities, Geophysics, 76 (2011), F215–F226.Google Scholar
[27]Namiki, T., A new FDTD algorithm based on alternating-direction implicit method, IEEE Trans. Microwave Theory Tech., 47 (1999), 2003–2007.CrossRefGoogle Scholar
[28]Oristaglio, M. L. and Hohmann, G. W., Diffusion of electromagnetic fields into a two-dimensional earth: A finite-difference approach, Geophysics, 49 (1984), 870–894.Google Scholar
[29]Peaceman, D. W. and Rachford, H. H. Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3 (1955), 28–41.Google Scholar
[30]Puzyrev, V., Koldan, J., de la Puente, J., Houzeaux, G., Vázquez, M. and Cela, J. M., A parallel finite-element method for three-dimensional controlled-source electromagnetic forward modelling, Geophys. J. Int., 193 (2013), 678–693.Google Scholar
[31]Ren, Z., Kalscheuer, T., Greenhalgh, S. and Maurer, H., A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling, Geophys. J. Int., 194 (2013), 700–718.Google Scholar
[32]Schwarzbach, C. and Haber, E., Finite element based inversion for time-harmonic electromagnetic problems, Geophys. J. Int., 193 (2013), 615–634.Google Scholar
[33]Sheu, W. H. T., Liang, L. Y. and Li, J.-H., Development of an explicit symplectic scheme that optimizes the dispersion-relation equation for the Maxwell's equations, Comm. Comput. Phys., 13 (2013), 1107–1133.Google Scholar
[34]Sommer, M., Hölz, S., Moorkamp, M., Swidinsky, A., Heincke, B., Scholl, C. and Jegen, M., GPU parallelization of a three dimensional marine CSEM code, Comput. Geosci., 58 (2013), 91–99.Google Scholar
[35]Streich, R., 3D finite-difference frequency-domain modeling of controlled-source electromagnetic data: Direct solution and optimization for high accuracy, Geophysics, 74 (2009), F95–F105.Google Scholar
[36]Streich, R., Becken, M. and Ritter, O., 2.5D controlled-source EM modeling with general 3D source geometries, Geophysics, 76 (2011), F387–F393.Google Scholar
[37]Sun, Y. and Tao, F., Symplectic and multisymplectic numerical methods for Maxwell's equations, J. Comput. Phys., 230 (2011), 2076–2094.CrossRefGoogle Scholar
[38]Taflove, A., Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems, IEEE Trans. Electromagn. Compat., 22 (1980), 191–202.Google Scholar
[39]Thomas, L., Elliptic problems in linear difference equations over a network, Technique report, Watson Scientific Computing Laboratory, Columbia University, New York, 1949.Google Scholar
[40]Um, E. S., Commer, M. and Newman, G. A., Efficient pre-conditioned iterative solution strategies for the electromagnetic diffusion in the earth: finite-element frequency-domain approach, Geophys. J. Int, 193 (2013), 1460–1473.Google Scholar
[41]Um, E. S., Harris, J. M. and Alumbaugh, D. L., 3D time-domain simulation of electromagnetic diffusion phenomena: A finite-element electric-field approach, Geophysics, 75 (2010), F115–F126.Google Scholar
[42]Wang, T. and Hohmann, G. W., A finite-difference, time-domain solution for three-dimensional electromagnetic modeling, Geophysics, 58 (1993), 797–809.Google Scholar
[43]Xie, Z., Wang, B. and Zhang, Z., Space-time discontinuous Galerkin method for Maxwell's equations, Comm. Comput. Phys., 14 (2013), 916–939.Google Scholar
[44]Yee, K. S.et al., Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media, IEEE Trans. Antennas Propag, 14 (1966), 302–307.Google Scholar
[45]Zaslavsky, M., Druskin, V., Davydycheva, S., Knizhnerman, L., Abubakar, A. and Habashy, T., Hybrid finite-difference integral equation solver for 3D frequency domain anisotropic electromagnetic problems, Geophysics, 76 (2011), F123–F137.Google Scholar