Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations

Jiangxing Wang

doi:10.3934/dcdsb.2020185

Article Contents

2021, Volume 26, Issue 5: 2429-2440. Doi: 10.3934/dcdsb.2020185

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Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations

Jiangxing Wang^1,2, ,

1.
MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China
2.
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore

^* Corresponding author: Jiangxing Wang
^* Corresponding author: Jiangxing Wang

Received: December 31, 2019

Revised: February 29, 2020

Early access: June 2020

Published: May 2021

This work was supported in part by the NSFC No.11801171, Hunan Provincial Natural Science Foundation of China(NO. 2019JJ50384), Scientific Research Found of Hunan Provincial Education Department(No.18B023)and Singapore MOE AcRF Tier 2 Grants: MOE2017-T2-2-144 and MOE2018-T2-1-059

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Abstract

In this paper, we investigate an accurate and efficient method for nonlinear Maxwell's equation. DG method and Crank-Nicolson scheme are employed for spatial and time discretization, respectively. A semi-explicit extrapolation technique is adopted for the linearization of the nonlinear term. Since the proposed scheme is semi-implicit, only a linear system needs to be solved at each time step. Optimal convergent order of $ O(\tau^2+h^{p+\frac{1}{2}}) $ is proved under time step size condition $ \tau\leq h^{d/4} $. Finally, 2D and 3D numerical examples are provided to validate the theoretical convergence rate.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. $ k = 1, \tau = h $ (Left), $ k = 2, \tau = h^2 $ (Right)

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Figure 2. $ p = 1, \tau = h $ (Left), $ p = 2, \tau = h^2 $ (Right)

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References

[1]	S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Springer Science & Business Media, 2007. doi: 10.1007/978-0-387-75934-0.
[2]	T. Chen, T. Kang, G. Lu and L. Wu, A $(T, \psi)-\psi_e$ decoupled scheme for a time-dependent multiply-connected eddy current problem, Math. Method. Appl. Sci., 37 (2014), 343-359. doi: 10.1002/mma.2795.
[3]	B. Cockburn, F. Li and C. W.-Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comp. Phys., 194 (2004), 488-610. doi: 10.1016/j.jcp.2003.09.007.
[4]	B. Cockburn and C. W.-Shu, Runge–Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comp., 16 (2001), 173-261. doi: 10.1023/A:1012873910884.
[5]	S. Durand and M. Slodicka, Fully discrete finite element method for Maxwell's equations with nonlinear conductivity, IMA J. Numer. Anal., 31 (2011), 1713-1733. doi: 10.1093/imanum/drr007.
[6]	S. Durand and M. Slodicka, Convergence of the mixed finite element method for Maxwell's equations with nonlinear conductivity, Math. Methods in the Applied Sciences, 35 (2012), 1489-1504. doi: 10.1002/mma.2513.
[7]	M. Ferreira and C. Buriol, Orthogonal decomposition and asymptotic behavior for nonlinear Maxwell's equations, J. Math. Anal. Appl., 426 (2015), 392-405. doi: 10.1016/j.jmaa.2014.12.071.
[8]	L. Fezoui, S. Lanteri, S. Lohrengel and S. Piperno, Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, ESAIM: Model. Math. Anal. Numer., 39 (2005), 1149-1176. doi: 10.1051/m2an:2005049.
[9]	G. Gruner, A. Zawadowski and P. Chaikin, Nonlinear conductivity and noise due to charge-density-wave depinning in nb se 3, Physical Review Letters, 46 (1981), 511-515.
[10]	R. Guo, L. Ji and Y. Xu, High order local local discontinuous Galerkin method for the Allen-Cahn equation: Analysis and simulation, J. Comput. Math., 34 (2016), 135-158. doi: 10.4208/jcm.1510-m2014-0002.
[11]	J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2008. doi: 10.1007/978-0-387-72067-8.
[12]	H. Jia, J. Li, Z. Fang and M. Li, A new FDTD scheme for Maxwell's equations in Kerr-type nonlinear media, Numerical Algorithms, 82 (2019), 223-243. doi: 10.1007/s11075-018-0602-3.
[13]	J. Li and J. S. Hesthaven, Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comp. Phys., 258 (2014), 915-930. doi: 10.1016/j.jcp.2013.11.018.
[14]	A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus Ltd., London, 1983.
[15]	T. Kang, Y. Wang, L. Wu and K. Kim, An improved error estimate for Maxwell's equations with a power-law nonlinear conductivity, Appl. Math. Lett., 45 (2015), 93-97. doi: 10.1016/j.aml.2015.01.017.
[16]	W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479, (1973).
[17]	J. Rhyner, Magnetic properties and AC-losses of superconductors with power law current voltage characteristics, Physica C: Superconductivity, 212 (1993), 292-300. doi: 10.1016/0921-4534(93)90592-E.
[18]	M. Slodicka, A time discretization scheme for a non-linear degenerate eddy current model for ferromagnetic materials, IMA J. Numer. Anal., 26 (2006), 173-186. doi: 10.1093/imanum/dri030.
[19]	M. Slodicka and S. Durand, Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition, J. Math. Anal. Appl., 375 (2011), 230-244. doi: 10.1016/j.jmaa.2010.09.016.
[20]	H. Song and C.-W. Shu, Uncodntional energy stability analysis of a second order implicit-explicit local discontinuous Galerkin method for the Cahn-Hilliard equation, J. Sci. Comput., 73 (2017), 1178-1203. doi: 10.1007/s10915-017-0497-5.
[21]	B. Wang, Z. Xie and Z. Zhang, Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media, J. Comput. Phys., 229 (2010), 8552-8563. doi: 10.1016/j.jcp.2010.07.038.
[22]	B. Wang, Z. Xie and Z. Zhang, Space-time discontinuous galerkin method for maxwell equations in dispersive media, Acta Math. Sci., 34 (2014), 1357-1376. doi: 10.1016/S0252-9602(14)60089-8.
[23]	J. Wang, Z. Xie and C. Chen, Implicit DG method for time domain Maxwell's equations involving metamaterials, Adv. Appl. Math. Mech., 7 (2015), 796-817. doi: 10.4208/aamm.2014.m725.
[24]	Z. Xie, J. Wang, C. Chen and B. Wang, Solving Maxwell's equation in meta-materials by a CG-DG method, Commun. Comput. Phys., 19 (2016), 1242-1264. doi: 10.4208/cicp.scpde14.35s.
[25]	C. Yao, Y. Lin, C. Wang and Y. Kou, A third order linearized BDFscheme for Maxwell's equations with nonlinear conductivity using finite element method, Int. J. Numer. Anal. Mod., 14 (2017), 511-531.
[26]	H. Yin, On a singular limit problem for nonlinear Maxwell's equations, J. Differential Equations, 156 (1999), 355-375. doi: 10.1006/jdeq.1998.3608.