论文标题
Dirac辅助树方法，用于具有任意可变波数的1D异质Helmholtz方程
Dirac Assisted Tree Method for 1D Heterogeneous Helmholtz Equations with Arbitrary Variable Wave Numbers
论文作者
论文摘要
在本文中，我们介绍了一种称为Dirac Assisted Tree（DAT）方法的新方法，该方法可以处理具有任意可变波数的1D异质Helmholtz方程。 DAT将原始的全局问题分解为许多并行树结构的小型本地问题，这些问题通过解决小型链接问题而链接在一起以形成全局解决方案。为了解决DAT中的局部问题，我们提出了一种紧凑的有限差异方法（FDM），该方法具有任意高精度顺序和低的数值分散，用于分段平滑系数和可变波数。这种紧凑的FDM对DAT特别有吸引力，因为局部问题及其在DAT中的通量可以很高的精度计算。具有这种紧凑的FDM的DAT可以通过求解小型线性系统来求解具有任意可变波数的异质Helmholtz方程 - 在极端情况下，$ 4 \ times 4 $矩阵 - 具有平行方式的Tridiagonal系数矩阵。提供了几个数值示例，以说明使用$ M $ M $ TH订单紧凑型FDM的有效性，其中$ M = 6,8 $用于数值求解具有可变波数的异质Helmholtz方程。我们还将讨论如何使用DAT解决一些特殊的2D Helmholtz方程。
In this paper we introduce a new method called the Dirac Assisted Tree (DAT) method, which can handle 1D heterogeneous Helmholtz equations with arbitrarily large variable wave numbers. DAT breaks an original global problem into many parallel tree-structured small local problems, which are linked together to form a global solution by solving small linking problems. To solve the local problems in DAT, we propose a compact finite difference method (FDM) with arbitrarily high accuracy order and low numerical dispersion for piecewise smooth coefficients and variable wave numbers. This compact FDM is particularly appealing for DAT, because the local problems and their fluxes in DAT can be computed with high accuracy. DAT with such compact FDMs can solve heterogeneous Helmholtz equations with arbitrarily large variable wave numbers accurately by solving small linear systems - $4 \times 4$ matrices in the extreme case - with tridiagonal coefficient matrices in a parallel fashion. Several numerical examples are provided to illustrate the effectiveness of DAT using the $M$th order compact FDMs with $M=6,8$ for numerically solving heterogeneous Helmholtz equations with variable wave numbers. We shall also discuss how to solve some special 2D Helmholtz equations using DAT.
