论文标题
通过完美匹配的层的一维非局部Helmholtz方程的数值解
Numerical solution of a one-dimensional nonlocal Helmholtz equation by Perfectly Matched Layers
论文作者
论文摘要
我们考虑使用完美匹配的层（PML）来考虑非本地Helmholtz方程的计算。我们首先通过将PML修改从本地运算符扩展到积分形式的非本地运算符来得出非局部PML方程。然后，我们对非本地Helmholtz解决方案的某些加权平均值进行稳定性估计，并证明（i）在一种情况下，非局部PML溶液的加权平均值在PML层中成倍衰减； （ii）在另一种情况下，非本地Helmholtz解决方案的加权平均值本身在某些域之外呈指数衰减。特别是对于典型的内核函数$γ_1（s）= \ frac12 e^{ - | s |} $，我们获得了非本地Helmholtz方程的绿色功能，并使用绿色的功能进一步证明（i）在一种情况下，非局部PML解决方案在PML层中呈指数衰减； （ii）在另一种情况下，非本地Helmholtz解决方案本身在某些域外呈指数衰减。基于我们的理论分析，讨论了截断的非局部问题，还引入了渐近兼容性方案以解决所得的截断问题。最后，提供了数值示例，以验证我们非本地PML策略和理论发现的有效性和验证。
We consider the computation of a nonlocal Helmholtz equation by using Perfectly Matched Layer (PML). We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form. We then give stability estimates of some weighted average value of the nonlocal Helmholtz solution and prove that (i) the weighted average value of the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the weighted average value of the nonlocal Helmholtz solution itself decays exponentially outside some domain. Particularly for a typical kernel function $γ_1(s)=\frac12 e^{-| s|}$, we obtain the Green's function of the nonlocal Helmholtz equation, and use the Green's function to further prove that (i) the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the nonlocal Helmholtz solution itself decays exponentially outside some domain. Based on our theoretical analysis, the truncated nonlocal problems are discussed and an asymptotic compatibility scheme is also introduced to solve the resulting truncated problems. Finally, numerical examples are provided to verify the effectiveness and validation of our nonlocal PML strategy and theoretical findings.
