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 Mth 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.

在本文中，我们介绍了一种称为狄拉克辅助树 (DAT) 方法的新方法，该方法可以处理具有任意大可变波数的一维异构亥姆霍兹方程。DAT 将一个原始的全局问题分解为许多并行的树状局部小问题，这些小问题通过解决小的连接问题链接在一起形成全局解决方案。为了解决 DAT 中的局部问题，我们针对分段平滑系数和可变波数提出了一种具有任意高精度阶数和低数值分散的紧凑有限差分法 (FDM)。这种紧凑的 FDM 对 DAT 尤其有吸引力，因为可以高精度计算 DAT 中的局部问题及其通量。$4\times 4$极端情况下的矩阵 - 以并行方式使用三对角系数矩阵。提供了几个数值例子来说明使用M阶紧凑 FDM的 DAT 的有效性$米=6,8$用于数值求解具有可变波数的异构亥姆霍兹方程。我们还将讨论如何使用 DAT 求解一些特殊的 2D Helmholtz 方程。