Positivity-preserving and unconditionally energy stable numerical schemes for MEMS model
Positivity-preserving and unconditionally energy stable numerical schemes for MEMS model
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DOI：
10.1016/j.apnum.2022.07.002
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发表时间：
2022-07
影响因子：
2.8
通讯作者：
Dianming Hou;Hui Wang;Chaowu Zhang
中科院分区：
文献类型：
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作者：
Dianming Hou;Hui Wang;Chaowu Zhang
本文提出并分析了微电子机械系统(MEMS)的保正性、无条件能量稳定和线性二阶全离散格式。更准确地说，我们使用一阶后向差分格式(BDF1)和二阶Crank-Nicolson(CN)格式进行时间离散，并使用中心有限差分方法进行空间离散。在我们的数值格式中采用了指数标量辅助变量(ESAV)方法的一个变体来处理奇异非线性项。严格证明了数值格式的无条件能量稳定性，对时间步长没有任何限制。此外，我们还推导出数值解在点水平上始终保持MEMS模型的正性，即距离变量始终在0和定常解之间。给出了一系列数值模拟，以证明我们的数值格式的正性和能量稳定性。
In this paper, we propose and analyze the positivity-preserving, unconditionally energy stable, and linear second order fully discrete schemes for Micro-Electromechanical system (MEMS). More precisely, we use the first order backward difference formulation (BDF1) and second order Crank-Nicolson (CN) formulation for the temporal discretization, and the central finite difference method for spatial discretization. A variant of the exponential scalar auxiliary variable (ESAV) approach is involved in our numerical schemes to deal with the singular nonlinear term. The unconditional energy stability of the numerical schemes is rigorously proved, without any restriction for the time step sizes. Furthermore, we derive that the numerical solutions always preserve the positivity property of the MEMS model, that is the distance variable is always between 0 and the steady solution, at a point-wise level. A series of numerical simulations are presented to demonstrate the positivity and energy stability of our numerical schemes.