Abstract
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 normsl2.
Highlights
The struggle for computing accurate solution using different grid sizes has increased researchers’ curiosity for developing high order difference schemes
It is shown that the sixth-order compact difference scheme is the most cost effective compared to the second-order finite-difference schemes and the corresponding conventional second-order central-difference scheme with Multigrid methods
We have studied a sixth-order compact finite-difference scheme with equal mesh sizes for discretizing a two and three-Dimensional Helmholtz equation
Summary
The struggle for computing accurate solution using different grid sizes has increased researchers’ curiosity for developing high order difference schemes. Singer and Turkel have conducted an important work in this regard [4] They used Dirichlet and/or Neumann boundary conditions for the development of a fourth-order compact finite-difference method. Later on a sixth-order finite-difference method was developed by Nabavi et al for solving Helmholtz equation in one-dimensional and two-dimensional domain with Neumann boundary conditions [7]. The present study is the first that uses sixth-order compact finite-difference scheme for three dimensions and the designing of specialized Multigrid method. This scheme is different from [1, 7] but the difference is of high order, because the term ∂6u/∂x2∂y2∂z2 in 3 dimensions has no counterpart in 2 dimensions. It is different when l is kept constant in [8]
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