Before revision:  All such local solutions are then linked together to form a global solution by solving small Dirac assisted linking problemswith 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.
After revision:  All such local solutions are then linked together to form a global solution by solving small linking problems. 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.