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Volume 14, Issue 4-5
A Third Order Linearized BDF Scheme for Maxwell's Equations with Nonlinear Conductivity Using Finite Element Method

Changhui Yao, Yanping Lin, Cheng Wang & Yanli Kou

Int. J. Numer. Anal. Mod., 14 (2017), pp. 511-531.

Published online: 2017-08

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  • Abstract

In this paper, we study a third order accurate linearized backward differential formula (BDF) type scheme for the nonlinear Maxwell's equations, using the Nédelec finite element approximation in space. A purely explicit treatment of the nonlinear term greatly simplifies the computational effort, since we only need to solve a constant-coefficient linear system at each time step. An optimal $L^2$ error estimate is presented, via a linearized stability analysis for the numerical error function, under a condition for the time step, $\tau \leq C^*_0h^2$ for a fixed constant $C^*_0$. Numerical results are provided to confirm our theoretical analysis and demonstrate the high order accuracy and stability (convergence) of the linearized BDF finite element method.

  • Keywords

Maxwell's equations with nonlinear conductivity, convergence analysis and optimal error estimate, linearized stability analysis, the third order BDF scheme.

  • AMS Subject Headings

35R35, 49J40, 60G40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-511, author = {Changhui Yao, Yanping Lin, Cheng Wang and Yanli Kou}, title = {A Third Order Linearized BDF Scheme for Maxwell's Equations with Nonlinear Conductivity Using Finite Element Method}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2017}, volume = {14}, number = {4-5}, pages = {511--531}, abstract = {

In this paper, we study a third order accurate linearized backward differential formula (BDF) type scheme for the nonlinear Maxwell's equations, using the Nédelec finite element approximation in space. A purely explicit treatment of the nonlinear term greatly simplifies the computational effort, since we only need to solve a constant-coefficient linear system at each time step. An optimal $L^2$ error estimate is presented, via a linearized stability analysis for the numerical error function, under a condition for the time step, $\tau \leq C^*_0h^2$ for a fixed constant $C^*_0$. Numerical results are provided to confirm our theoretical analysis and demonstrate the high order accuracy and stability (convergence) of the linearized BDF finite element method.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10047.html} }
TY - JOUR T1 - A Third Order Linearized BDF Scheme for Maxwell's Equations with Nonlinear Conductivity Using Finite Element Method AU - Changhui Yao, Yanping Lin, Cheng Wang & Yanli Kou JO - International Journal of Numerical Analysis and Modeling VL - 4-5 SP - 511 EP - 531 PY - 2017 DA - 2017/08 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10047.html KW - Maxwell's equations with nonlinear conductivity, convergence analysis and optimal error estimate, linearized stability analysis, the third order BDF scheme. AB -

In this paper, we study a third order accurate linearized backward differential formula (BDF) type scheme for the nonlinear Maxwell's equations, using the Nédelec finite element approximation in space. A purely explicit treatment of the nonlinear term greatly simplifies the computational effort, since we only need to solve a constant-coefficient linear system at each time step. An optimal $L^2$ error estimate is presented, via a linearized stability analysis for the numerical error function, under a condition for the time step, $\tau \leq C^*_0h^2$ for a fixed constant $C^*_0$. Numerical results are provided to confirm our theoretical analysis and demonstrate the high order accuracy and stability (convergence) of the linearized BDF finite element method.

Changhui Yao, Yanping Lin, Cheng Wang and Yanli Kou. (2017). A Third Order Linearized BDF Scheme for Maxwell's Equations with Nonlinear Conductivity Using Finite Element Method. International Journal of Numerical Analysis and Modeling. 14 (4-5). 511-531. doi:
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