High order ADI method for separable generalized Helmholtz equations

We present a multilevel high order ADI method for separable generalized Helmholtz equations. The discretization method we use is a one-dimensional fourth order compact finite difference applied to each directional component of the Laplace operator, resulting in a discrete system efficiently solvable by ADI methods. We apply this high order difference scheme to all levels of grids, and then starting from the coarsest grid, solve the discretized equation with an ADI method at each grid level, with the solution from the previous grid level as the initial guess. The multilevel procedure stops as the ADI finishes its iterations on the finest grid. Analytical and experimental results show that the proposed method is highly accurate and efficient while remaining as algorithmically and data-structurally simple as the single grid ADI method.