## Abstract and Applied Analysis

### Multigrid Method for Solution of 3D Helmholtz Equation Based on HOC Schemes

#### 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 norms ${l}_{2}$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 954658, 14 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425049589

Digital Object Identifier
doi:10.1155/2014/954658

Mathematical Reviews number (MathSciNet)
MR3256268

Zentralblatt MATH identifier
07023394

#### Citation

Ghaffar, Fazal; Badshah, Noor; Islam, Saeed. Multigrid Method for Solution of 3D Helmholtz Equation Based on HOC Schemes. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 954658, 14 pages. doi:10.1155/2014/954658. https://projecteuclid.org/euclid.aaa/1425049589

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