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, are linked together to form solution by solving linking problems. To solve problems in DAT, propose compact finite difference (FDM) high accuracy order and low numerical dispersion for piecewise smooth coefficients This FDM is particularly appealing because their fluxes be computed accuracy. such FDMs numbers accurately linear systems - $4 \times 4$ matrices extreme case tridiagonal coefficient fashion. Several examples provided illustrate effectiveness of using $M$th $M=6,8$ numerically We shall also discuss how some special 2D DAT.