High-order approximation compact schemes for forward subsurface scattering problems

Abstract

We develop an efficient iterative approach to the solution of the discrete three-dimensional Helmholtz equation with variable coefficients and PML boundary conditions based on compact fourth and sixth order approximation schemes. The coefficient matrices of the resulting systems are not Hermitian and possess positive as well as negative eigenvalues so represent a significant challenge for constructing an efficient iterative solver. In our approach these systems are solved by a combination of a Krylov subspace-type method with a matching high order approximation preconditioner with coefficients depending only on one spatial variable. In the algorithms considered, the direct solution of high order preconditioning system is based on a combination of the separation of variables technique and Fast Fourier Transform (FFT) type methods. The resulting numerical methods allow for efficient implementation on parallel computers. Numerical results confirm the high efficiency of the proposed iterative algorithms.