A Non-Stiff Summation-By-Parts Finite Difference Method for the Scalar Wave Equation in Second Order Form: Characteristic Boundary Conditions and Nonlinear Interfaces
A Non-Stiff Summation-By-Parts Finite Difference Method for the Scalar Wave Equation in Second Order Form: Characteristic Boundary Conditions and Nonlinear Interfaces
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DOI：
10.1007/s10915-022-01961-1
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发表时间：
2019-07
影响因子：
2.5
通讯作者：
B. Erickson;J. Kozdon;Tobias W. Harvey
中科院分区：
文献类型：
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作者：
B. Erickson;J. Kozdon;Tobias W. Harvey
曲线型多块逐段求和有限差分算子和同时逼近项方法为求解二阶波动方程提供了一个稳定而准确的框架。也就是说，当使用特征边界条件和非线性界面条件时，标准方法可能会变得任意僵硬。在这里，我们提出了一种新的技术，通过使用特征变量来“迎风”处理边界和界面来避免这种刚性。这是通过引入额外的块边界位移变量来实现的。对于标量各向异性波动方程，我们用统一的能量表示标准边界和特征边界以及界面处理，证明了所得到的格式具有半离散能量稳定性。数值实验验证了理论上的稳定性结果，也证明了该方法的准确性和稳健性。数值结果还表明，特征格式具有基于标准波传播考虑的时间步长限制，而不是基于边界闭合。
Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard method can become arbitrarily stiff when characteristic boundary conditions and nonlinear interface conditions are used. Here we propose a new technique that avoids this stiffness by using characteristic variables to “upwind” the boundary and interface treatment. This is done through the introduction of an additional block boundary displacement variable. Using a unified energy, which expresses both the standard as well as characteristic boundary and interface treatment, we show that the resulting scheme has semidiscrete energy stability for the scalar anisotropic wave equation. The theoretical stability results are confirmed with numerical experiments that also demonstrate the accuracy and robustness of the proposed scheme. The numerical results also show that the characteristic scheme has a time step restriction based on standard wave propagation considerations and not the boundary closure.