Higher order rotated iterative scheme for the 2D Helmholtz equation

Abstract

Improved techniques derived from the rotated finite difference operators have been developed over the last few years in solving the linear systems that arise from the discretization of various partial differential equations (PDEs). Furthermore, a higher order system can be generated from discretization of the finite difference scheme using the fourth order compact scheme generated from the second order central difference. By using compact finite differences, a new rotated point scheme with fourth-order accuracy for the two-dimensional (2D) Helmholtz equation is formed. On the other hand, the multiscale multigrid method combined with Richardson's extrapolation is first introduced by Zhang to solve the 2D Poisson equation. By combining the fourth-order rotated scheme and multiscale multigrid method with Richardson's extrapolation in the solution of the 2D Helmholtz equation, the order of accuracy of the approximation can be improved up to sixth order, and with larger mesh size, the convergence rate of these iterative methods is faster as well. Numerical experiments are conducted on the rotated scheme combined with multiscale multigrid method and Richardson's extrapolation, and the result is compared with existing point methods and multigrid method. The results show the improvements in the convergence rate and the efficiency of the newly formulated iterative scheme.