Parallel solution of high frequency Helmholtz equations using high order finite difference schemes

Abstract

High frequency Helmholtz equations pose a challenging computational task, especially at high frequencies. This work examines the parallel solution of such problems, using 2nd, 4th and 6th order finite difference schemes. The examples include problems with known analytic solutions, enabling error evaluation of the different schemes on various grids, including problems on L-shaped domains. The resulting linear systems are solved with the block-parallel CARP-CG algorithm which succeeded in lowering the relative residual, indicating that it is a robust and reliable parallel solver of the resulting linear systems. However, lowering the error of the solution to reasonable levels with practical mesh sizes is obtained only with the higher order schemes. These results corroborate the known limitations of the 2nd order scheme at modeling the Helmholtz equation at high frequencies, and they indicate that CARP-CG can also be used effectively with high order finite difference schemes. Furthermore, the parallel scalability of CARP-CG improves when the wave number is increased (on a fixed grid), or when, for a fixed wave number, the grid size is decreased.