November  2018, 38(11): 5685-5709. doi: 10.3934/dcds.2018248

Error analysis of an ADI splitting scheme for the inhomogeneous Maxwell equations

Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

* Corresponding author: Roland Schnaubelt

Received  December 2017 Revised  June 2018 Published  August 2018

Fund Project: The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173

In this paper we investigate an alternating direction implicit (ADI) time integration scheme for the linear Maxwell equations with currents, charges and conductivity. We show its stability and efficiency. The main results establish that the scheme converges in a space similar to $H^{-1}$ with order two to the solution of the Maxwell system. Moreover, the divergence conditions in the system are preserved in $H^{-1}$ with order one.

Citation: Johannes Eilinghoff, Roland Schnaubelt. Error analysis of an ADI splitting scheme for the inhomogeneous Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5685-5709. doi: 10.3934/dcds.2018248
References:
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R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier, Amsterdam, 2003. Google Scholar

[2]

H. Amann, Linear and Quasilinear Parabolic Problems. Volume Ⅰ: Abstract Linear Theory Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9221-6. Google Scholar

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C. AmroucheC. BernardiM. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823-864. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B. Google Scholar

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W. ChenX. Li and D. Liang, Energy-conserved splitting FDTD methods for Maxwell's equations, Numer. Math., 108 (2008), 445-485. doi: 10.1007/s00211-007-0123-9. Google Scholar

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W. ChenX. Li and D. Liang, Energy-conserved splitting finite-difference time-domain methods for Maxwell's equations in three dimensions, SIAM J. Numer. Anal., 48 (2010), 1530-1554. doi: 10.1137/090765857. Google Scholar

[6]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 3: Spectral Theory and Applications, Springer, Berlin, 1990. Google Scholar

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R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 5: Evolution Problems I, Springer, Berlin, 1992. doi: 10.1007/978-3-642-58090-1. Google Scholar

[8]

J. Eilinghoff, Error estimates of splitting methods for wave type equations, Ph. D. thesis, Karlsruhe, 2017, see https://publikationen.bibliothek.kit.edu/1000075070.Google Scholar

[9]

J. Eilinghoff and R. Schnaubelt, Error estimates in L2 of an ADI splitting scheme for the inhomogeneous Maxwell, preprint, see http://www.math.kit.edu/iana3/$\sim$schnaubelt/media/adi-strong.pdf.Google Scholar

[10]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. Google Scholar

[11]

L. GaoB. Zhang and D. Liang, The splitting finite-difference time-domain methods for Maxwell's equations in two dimensions, J. Comput. Appl. Math., 205 (2007), 207-230. doi: 10.1016/j.cam.2006.04.051. Google Scholar

[12]

E. Hansen and A. Ostermann, Dimension splitting for evolution equations, Numer. Math., 108 (2008), 557-570. doi: 10.1007/s00211-007-0129-3. Google Scholar

[13]

M. HochbruckT. Jahnke and R. Schnaubelt, Convergence of an ADI splitting for Maxwell's equations, Numer. Math., 129 (2015), 535-561. doi: 10.1007/s00211-014-0642-0. Google Scholar

[14]

M. Hochbruck and A. Sturm, Error analysis of a second-order locally implicit method for linear Maxwell's equations, SIAM J. Numer. Anal., 54 (2016), 3167-3191. doi: 10.1137/15M1038037. Google Scholar

[15]

M. Hochbruck and A. Sturm, Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell's equations preprint 2017/12 of CRC 1172, see http://www.waves.kit.edu/downloads/CRC1173_Preprint_2017-12.pdf. doi: 10.1090/mcom/3365. Google Scholar

[16]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. Google Scholar

[17]

P. C. Kunstmann and L. Weis, Maximal Lp-regularity for parabolic equations, Fourier multiplier theorems and H-functional calculus, In Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, R. Nagel and S. Piazzera), Springer-Verlag, 1855 (2004), 65-311. doi: 10.1007/978-3-540-44653-8_2. Google Scholar

[18]

J. Lee and B. Fornberg, A split step approach for the 3-D Maxwell's equations, J. Comput. Appl. Math., 158 (2003), 485-505. doi: 10.1016/S0377-0427(03)00484-9. Google Scholar

[19]

A. Lunardi, Interpolation Theory, Edizione della Normale, Pisa, 2009. Google Scholar

[20]

T. Namiki, 3-D ADI-FDTD method-unconditionally stable time-domain algorithm for solving full vector Maxwell's equations, IEEE Trans. Microwave Theory Tech., 48 (2000), 1743-1748. Google Scholar

[21]

J. Nečas, Direct Methods in the Theory of Elliptic Equations Springer-Verlag, Heidelberg, 2012.Google Scholar

[22]

A. Ostermann and K. Schratz, Error analysis of splitting methods for inhomogeneous evolution equations, Appl. Numer. Math., 62 (2012), 1436-1446. doi: 10.1016/j.apnum.2012.06.002. Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[24]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Second edition. With 1 CD-ROM (Windows). Artech House, Inc., Boston, MA, 2000. Google Scholar

[25]

K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Trans. Antennas Propagation, 14 (1966), 302-307. Google Scholar

[26]

F. ZhengZ. Chen and J. Zhang, Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method, IEEE Trans. Microwave Theory Tech., 48 (2000), 1550-1558. Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier, Amsterdam, 2003. Google Scholar

[2]

H. Amann, Linear and Quasilinear Parabolic Problems. Volume Ⅰ: Abstract Linear Theory Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9221-6. Google Scholar

[3]

C. AmroucheC. BernardiM. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823-864. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B. Google Scholar

[4]

W. ChenX. Li and D. Liang, Energy-conserved splitting FDTD methods for Maxwell's equations, Numer. Math., 108 (2008), 445-485. doi: 10.1007/s00211-007-0123-9. Google Scholar

[5]

W. ChenX. Li and D. Liang, Energy-conserved splitting finite-difference time-domain methods for Maxwell's equations in three dimensions, SIAM J. Numer. Anal., 48 (2010), 1530-1554. doi: 10.1137/090765857. Google Scholar

[6]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 3: Spectral Theory and Applications, Springer, Berlin, 1990. Google Scholar

[7]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 5: Evolution Problems I, Springer, Berlin, 1992. doi: 10.1007/978-3-642-58090-1. Google Scholar

[8]

J. Eilinghoff, Error estimates of splitting methods for wave type equations, Ph. D. thesis, Karlsruhe, 2017, see https://publikationen.bibliothek.kit.edu/1000075070.Google Scholar

[9]

J. Eilinghoff and R. Schnaubelt, Error estimates in L2 of an ADI splitting scheme for the inhomogeneous Maxwell, preprint, see http://www.math.kit.edu/iana3/$\sim$schnaubelt/media/adi-strong.pdf.Google Scholar

[10]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. Google Scholar

[11]

L. GaoB. Zhang and D. Liang, The splitting finite-difference time-domain methods for Maxwell's equations in two dimensions, J. Comput. Appl. Math., 205 (2007), 207-230. doi: 10.1016/j.cam.2006.04.051. Google Scholar

[12]

E. Hansen and A. Ostermann, Dimension splitting for evolution equations, Numer. Math., 108 (2008), 557-570. doi: 10.1007/s00211-007-0129-3. Google Scholar

[13]

M. HochbruckT. Jahnke and R. Schnaubelt, Convergence of an ADI splitting for Maxwell's equations, Numer. Math., 129 (2015), 535-561. doi: 10.1007/s00211-014-0642-0. Google Scholar

[14]

M. Hochbruck and A. Sturm, Error analysis of a second-order locally implicit method for linear Maxwell's equations, SIAM J. Numer. Anal., 54 (2016), 3167-3191. doi: 10.1137/15M1038037. Google Scholar

[15]

M. Hochbruck and A. Sturm, Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell's equations preprint 2017/12 of CRC 1172, see http://www.waves.kit.edu/downloads/CRC1173_Preprint_2017-12.pdf. doi: 10.1090/mcom/3365. Google Scholar

[16]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. Google Scholar

[17]

P. C. Kunstmann and L. Weis, Maximal Lp-regularity for parabolic equations, Fourier multiplier theorems and H-functional calculus, In Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, R. Nagel and S. Piazzera), Springer-Verlag, 1855 (2004), 65-311. doi: 10.1007/978-3-540-44653-8_2. Google Scholar

[18]

J. Lee and B. Fornberg, A split step approach for the 3-D Maxwell's equations, J. Comput. Appl. Math., 158 (2003), 485-505. doi: 10.1016/S0377-0427(03)00484-9. Google Scholar

[19]

A. Lunardi, Interpolation Theory, Edizione della Normale, Pisa, 2009. Google Scholar

[20]

T. Namiki, 3-D ADI-FDTD method-unconditionally stable time-domain algorithm for solving full vector Maxwell's equations, IEEE Trans. Microwave Theory Tech., 48 (2000), 1743-1748. Google Scholar

[21]

J. Nečas, Direct Methods in the Theory of Elliptic Equations Springer-Verlag, Heidelberg, 2012.Google Scholar

[22]

A. Ostermann and K. Schratz, Error analysis of splitting methods for inhomogeneous evolution equations, Appl. Numer. Math., 62 (2012), 1436-1446. doi: 10.1016/j.apnum.2012.06.002. Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[24]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Second edition. With 1 CD-ROM (Windows). Artech House, Inc., Boston, MA, 2000. Google Scholar

[25]

K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Trans. Antennas Propagation, 14 (1966), 302-307. Google Scholar

[26]

F. ZhengZ. Chen and J. Zhang, Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method, IEEE Trans. Microwave Theory Tech., 48 (2000), 1550-1558. Google Scholar

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