Development and analysis of two new finite element schemes for a time-domain carpet cloak model
Development and analysis of two new finite element schemes for a time-domain carpet cloak model
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DOI：
10.1007/s10444-022-09948-0
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发表时间：
2022-04
影响因子：
1.7
通讯作者：
Jichun Li;Chi-Wang Shu
中科院分区：
文献类型：
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作者：
Jichun Li;Chi-Wang Shu
在本文中，我们关注的是时域地毯斗篷模型，该模型最初是在我们先前的工作Li等人（SIAM J. Appl. Math.，第74卷第4期，第74页。1136-1151，）。在Li等人（SIAM J. Appl. Math.，74（4），pp. 1136-1151）和Li等人（Methods Appl. Math.，19（2），pp. 359-378，）。然而，这些显式格式的数值稳定性仅在时间步长τ=O（h ~ 2）的约束下得到了证明，这是不切实际的，而且限制太多。为了克服这一缺点，我们提出了两个新的有限元格式来求解这个地毯斗篷模型：一个是隐式Crank-Nicolson（CN）格式，另一个是显式蛙跳（LF）格式。受连续模型的一种新能量的启发，我们证明了CN格式在CFL约束τ=O（h）下的无条件稳定性和LF格式的条件稳定性.这两种数值稳定性都继承了连续稳定性的精确形式。最佳误差估计也建立了LF计划。最后，使用LF方案的数值结果来支持我们的分析，并证明了隐身现象。
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, ). 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, ) and Li et al. (Methods Appl. Math.,19(2), pp. 359–378, ). However, numerical stabilities for those proposed explicit schemes are only proved under the time step constrainτ=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 constraintτ=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.