A higher order compact difference (HOC) scheme with uniform mesh sizes in different coordinate directions is employed to discretize a two- and three-dimensional Helmholtz equation. In case of two dimension, the stencil is of 9 points while in three-dimensional case, the scheme has 27 points and has fourth- to fifth-order accuracy. Multigrid method using Gauss-Seidel relaxation is designed to solve the resulting sparse linear systems. Numerical experiments were conducted to test the accuracy of the sixth-order compact difference scheme with Multigrid method and to compare it with the standard second-order finite-difference scheme and fourth-order compact difference scheme. Performance of the scheme is tested through numerical examples. Accuracy and efficiency of the new scheme are established by using the errors norms .
Fazal Ghaffar. Noor Badshah. Saeed Islam. "Multigrid Method for Solution of 3D Helmholtz Equation Based on HOC Schemes." Abstr. Appl. Anal. 2014 (SI51) 1 - 14, 2014. https://doi.org/10.1155/2014/954658