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

  • * Corresponding author: Jiangxing Wang

    * Corresponding author: Jiangxing Wang
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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  • 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.

    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)

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

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