Dirac Assisted Tree Method for 1D Heterogeneous Helmholtz Equations with Arbitrary Variable Wave Numbers

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.