A new finite difference scheme for the 3D Helmholtz equation with a preconditioned iterative solver

摘要

In this paper, we propose a new finite difference scheme for the 3D Helmhoitz problem, which is compact and fourth-order in accuracy. Different from a standard compact fourth-order one, the new scheme is specially established based on minimizing the numerical dispersion, by approximating the zeroth-order term of the equation with a weighted-average for the values at 27 points. To determine optimal weight parameters, an optimization problem is formulated and then dealt with the singular value decomposition method based on the dispersion equation. For the proposed scheme, by skillfully splitting the 3D error equation into several 1D difference problems, the solution's uniqueness and convergence are derived with an effort. To solve the resulting linear system stemming from difference discretization, which is sparse and large-sized, we develop a Bi-CGSTAB iterative solver based on the preconditioning of shifted-laplacian and 3D full-coarsening multigrid. The shifted-laplacian is used to generate the preconditioner with a discretization by the proposed compact fourth-order scheme, while the full-coarsening multigrid with matrix-based prolongation operators is built to approximate the inverse of the preconditioner. Finally, numerical examples are presented to demonstrate the efficiency of the new difference scheme and the preconditioned solver.