A space-time spectral method for the 1-D Maxwell equation

Hui-qing Liao; Ying Fu; He-ping Ma; Hui-qing Liao; Ying Fu; He-ping Ma

doi:10.3934/math.2021444

AIMS Mathematics

2021, Volume 6, Issue 7: 7649-7668. doi: 10.3934/math.2021444

Previous Article Next Article

Research article

A space-time spectral method for the 1-D Maxwell equation

Department of Mathematics, Shanghai University, Shanghai, 200444, China

Received: 13 January 2021 Accepted: 22 April 2021 Published: 12 May 2021
MSC : 65M12, 65M70

A Legendre-tau space-time spectral method is established for the 1-D Maxwell equation. The polynomials of different degrees are used to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. Also, the time multi-interval Legendre-tau space-time spectral method is considered to keep the long-time computation stable. Error estimates for the method of single and multi-internal are given, respectively. Moreover, the space-time spectral method is applied to the numerical solutions of the 1-D nonlinear Maxwell equation and describes its implicit-explicit iteration scheme. Numerical examples are compared with some other methods, which verifies the effectiveness of the methods for the 1-D Maxwell equation.
- Maxwell equation,
- space-time spectral method,
- Legendre polynomial,
- interval decomposition,
- error estimate
Citation: Hui-qing Liao, Ying Fu, He-ping Ma. A space-time spectral method for the 1-D Maxwell equation[J]. AIMS Mathematics, 2021, 6(7): 7649-7668. doi: 10.3934/math.2021444

Related Papers:

Abstract

A Legendre-tau space-time spectral method is established for the 1-D Maxwell equation. The polynomials of different degrees are used to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. Also, the time multi-interval Legendre-tau space-time spectral method is considered to keep the long-time computation stable. Error estimates for the method of single and multi-internal are given, respectively. Moreover, the space-time spectral method is applied to the numerical solutions of the 1-D nonlinear Maxwell equation and describes its implicit-explicit iteration scheme. Numerical examples are compared with some other methods, which verifies the effectiveness of the methods for the 1-D Maxwell equation.

References

[1]	Y. Xu, C. W. Shu, Optimal error estimates of the semidiscrete local discontinuous Galerkin methods for high order wave equations, SIAM J. Numer. Anal., 50 (2012), 79-104. doi: 10.1137/11082258X
[2]	Y. Xu, C. W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys., 7 (2010), 1-46.
[3]	B. Dong, C. W. Shu, Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems, SIAM J. Numer. Anal., 47 (2009), 3240-3268. doi: 10.1137/080737472
[4]	H. J. Wang, C. W. Shu, Q. Zhang, Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems, SIAM J. Numer. Anal., 53 (2015), 206-227. doi: 10.1137/140956750
[5]	H. J. Wang, S. P. Wang, Q. Zhang, C. W. Shu, Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems, ESAIM: M2AN, 50 (2016), 1083-1105. doi: 10.1051/m2an/2015068
[6]	H. J. Wang, Y. X. Liu, Q. Zhang, C. W. Shu, Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow, Math. Comp., 88 (2019), 91-121.
[7]	B. Y. Guo, Z. Q. Wang, Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comput. Math., 30 (2009), 249-280. doi: 10.1007/s10444-008-9067-6
[8]	C. T. Sheng, Z. Q. Wang, B. Y. Guo, A multistep Legendre-Gauss spectral collocation method for nonlinear Volterra integral equations, SIAM J. Numer. Anal., 52 (2014), 1953-1980. doi: 10.1137/130915200
[9]	B. Y. Guo, Z. Q. Wang, A spectral collocation method for solving initial value problems of first order ordinary differential equations, DCDS-B, 14 (2010), 1029-1054. doi: 10.3934/dcdsb.2010.14.1029
[10]	J. G. Tang, H. P. Ma, Single and multi-interval Legendre spectral methods in time for parabolic equations, Numer. Meth. Part. D. E., 22 (2006), 1007-1034. doi: 10.1002/num.20135
[11]	T. A. Driscoll, B. Fornberg, A block pseudospectral method for Maxwell's equations. I. One-dimensional case, J. Comput. Phys., 140 (1998), 47-65. doi: 10.1006/jcph.1998.5883
[12]	T. A. Driscoll, B. Fornberg, Block pseudospectral methods for Maxwell's equations. II. Two-dimensional, discontinuous-coefficient case, SIAM J. Sci. Comput., 21 (1999), 1146-1167. doi: 10.1137/S106482759833320X
[13]	T. Namiki, A new fdtd algorithm based on alternating-direction implicit method, IEEE T. Microw. Theory, 47 (1999), 2003-2007. doi: 10.1109/22.795075
[14]	K. Yee, Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media, IEEE T. Antenn. Propag., 14 (1996), 302-307.
[15]	W. B. Chen, X. J. Li, D. Liang, Energy-conserved splitting FDTD methods for Maxwell's equations, Numer. Math., 108 (2008), 445-485. doi: 10.1007/s00211-007-0123-9
[16]	W. B. Chen, X. J. Li, 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
[17]	F. H. Zeng, H. P. Ma, D. Liang, Energy-conserved splitting spectral methods for two dimensional Maxwell's equations, J. Comput. Appl. Math., 265 (2014), 301-321. doi: 10.1016/j.cam.2013.09.048
[18]	H. P. Ma, Y. H. Qin, Q. Ou, Multidomain Legendre-Galerkin Chebyshev-collocation method for one-dimensional evolution equations with discontinuity, Appl. Numer. Math., 111 (2017), 246-259. doi: 10.1016/j.apnum.2016.09.010
[19]	D. D. Fang, H. P. Ma, Multidomain Legendre tau method for the 1-D Maxwell equation with discontinuous solutions, J. Numer. Methods Comput. Appl., 39 (2018), 288-298.
[20]	S. Zhao, G. W. Wei, High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces, J. Comput. Phys., 200 (2004), 60-103. doi: 10.1016/j.jcp.2004.03.008
[21]	J. H. Xie, L. J. Yi, An $h$-$p$ version of the continuous Petrov-Galerkin time stepping method for nonlinear second-order delay differential equations, Appl. Numer. Math., 143 (2019), 1-19.
[22]	Y. C. Wei, L. J. Yi, An $hp$-version of the $C^0$-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems, Adv. Comput. Math., 46 (2020), 56. doi: 10.1007/s10444-020-09800-3
[23]	Thomas, J. W, Numerical partial differential equations: finite difference methods, New York: Springer-Verlag, 1995,261-360.
[24]	C. Bernardi, Y. Maday, Spectral methods, In: Handbook of numerical analysis, 5 (1997), 209-485.
[25]	Y. H. Qin, H. P. Ma, Legendre-tau-Galerkin and spectral collocation method for nonlinear evolution equations, Appl. Numer. Math., 153 (2020), 52-65. doi: 10.1016/j.apnum.2020.02.001
[26]	C. H. Yao, Y. P. Lin, C. Wang, Y. L. Kou, A third order linearized BDF scheme for Maxwell's equations with nonlinear conductivity using finite element method, Int. J. Numer. Anal. Model., 14 (2017), 511-531.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)