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

In this paper we introduce a method called Dirac Assisted Tree (DAT), which can handle heterogeneous Helmholtz equations with arbitrarily large variable wave numbers. DAT breaks an original global problem into many parallel tree-structured small local problems, which can be effectively solved. All such local solutions are then linked together to form a global solution by solving small Dirac assisted linking problems with an inherent tree structure. DAT is embedded with the following attractive features: domain decomposition for reducing the problem size, tree structure and tridiagonal matrices for computational efficiency, and adaptivity for further improved performance. In order to solve the local problems in DAT, we shall propose a compact finite difference scheme with arbitrarily high accuracy order and low numerical dispersion for piecewise smooth coefficients and variable wave numbers. Such schemes are particularly appealing for DAT, because the local problems and their fluxes in DAT can be computed with high accuracy. With the aid of such high-order compact finite difference schemes, DAT can solve heterogeneous Helmholtz equations with arbitrarily large variable wave numbers accurately by solving small linear systems - 4 by 4 matrices in the extreme case - with tridiagonal coefficient matrices in a parallel fashion. Several examples will be provided to illustrate the effectiveness of DAT and compact finite difference schemes in numerically solving heterogeneous Helmholtz equations with variable wave numbers. We shall also discuss how to solve some special two-dimensional Helmholtz equations using DAT developed for one-dimensional problems. As demonstrated in all our numerical experiments, the convergence rates of DAT measured in relative $L_2$, $L_\infty$ and $H^1$ energy norms as a result of using our M-th order compact finite difference scheme are of order M.