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Development and analysis of two new finite element schemes for a time-domain carpet cloak model

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Abstract

In this paper, we are concerned about a time-domain carpet cloak model, which was originally derived in our previous work Li et al. (SIAM J. Appl. Math., 74(4), pp. 1136–1151, 2014). Some finite element schemes have been developed for this model and used to simulate the cloaking phenomenon in Li et al. (SIAM J. Appl. Math., 74(4), pp. 1136–1151, 2014) and Li et al. (Methods Appl. Math., 19(2), pp. 359–378, 2019). However, numerical stabilities for those proposed explicit schemes are only proved under the time step constrain τ = O(h2), which is impractical and too restricted. To overcome this disadvantage, we propose two new finite element schemes for solving this carpet cloak model: one is the implicit Crank-Nicolson (CN) scheme, and another one is the explicit leap-frog (LF) scheme. Inspired by a totally new energy developed for the continuous model, we prove the unconditional stability for the CN scheme and conditional stability for the LF scheme under the usual CFL constraint τ = O(h). Both numerical stabilities inherit the exact form as the continuous stability. Optimal error estimate is also established for the LF scheme. Finally, numerical results using the LF scheme are presented to support our analysis and demonstrate the cloaking phenomenon.

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Acknowledgements

J. Li would like to thank UNLV for granting his sabbatical leave during Spring 2021 so that he could have time working on this paper. We are grateful for two anonymous reviewers for their sightful comments on improving our paper.

Funding

J. Li’s work is supported by NSF grant DMS-2011943. C.-W. Shu’s work is supported by NSF grant DMS-2010107 and AFOSR grant FA9550-20-1-0055. W. Yang’s work is supported by Project of Scientifific Research Fund of Hunan Provincial Science and Technology department (No. 2018WK4006), NSFC Projects 12171411 and 11771371.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, 89154-4020, USA

    Jichun Li

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Chi-Wang Shu

  3. Hunan Key Laboratory for Computation and Simulation in Science Engineering, Xiangtan University, Xiangtan, China

    Wei Yang

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  1. Jichun Li
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Correspondence to Jichun Li.

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Li, J., Shu, CW. & Yang, W. Development and analysis of two new finite element schemes for a time-domain carpet cloak model. Adv Comput Math 48, 24 (2022). https://doi.org/10.1007/s10444-022-09948-0

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