American Institute of Mathematical Sciences

Advanced
May  2021, 26(5): 2429-2440. doi: 10.3934/dcdsb.2020185

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

Received  January 2020 Revised  March 2020 Published  June 2020

Fund Project: 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

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.

Keywords: Discontinuous Galerkin method, accurate and efficient, convergence analysis, fully discrete.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2008. doi: 10.1007/978-0-387-72067-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus Ltd., London, 1983. Google Scholar

[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.  Google Scholar

[16]

W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479, (1973). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2008. doi: 10.1007/978-0-387-72067-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus Ltd., London, 1983. Google Scholar

[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.  Google Scholar

[16]

W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479, (1973). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040
[2]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[3]

Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1863-1886. doi: 10.3934/jimo.2020050
[4]

Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021077
[5]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[6]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[7]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[8]

Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030
[9]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[10]

Zehui Jia, Xue Gao, Xingju Cai, Deren Han. The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1943-1971. doi: 10.3934/jimo.2020053
[11]

Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080
[12]

Tadeusz Kaczorek, Andrzej Ruszewski. Analysis of the fractional descriptor discrete-time linear systems by the use of the shuffle algorithm. Journal of Computational Dynamics, 2021  doi: 10.3934/jcd.2021007
[13]

Cheng Wang. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021019
[14]

Haiyan Wang, Jinyan Fan. Convergence properties of inexact Levenberg-Marquardt method under Hölderian local error bound. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2265-2275. doi: 10.3934/jimo.2020068
[15]

Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019
[16]

Françoise Demengel. Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3465-3488. doi: 10.3934/dcds.2021004
[17]

Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021, 16 (2) : 187-219. doi: 10.3934/nhm.2021004
[18]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200
[19]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395
[20]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, 2021, 15 (3) : 387-413. doi: 10.3934/ipi.2020073

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (99)
  • HTML views (308)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences