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**Multigrid method for solution of 3D Helmholtz equation based on HOC schemes.**
*(English)*
Zbl 1474.65420

Summary: 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 \(l_2\).

### MSC:

65N06 | Finite difference methods for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

### Software:

Wesseling
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\textit{F. Ghaffar} et al., Abstr. Appl. Anal. 2014, Article ID 954658, 14 p. (2014; Zbl 1474.65420)

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### References:

[1] | Sutmann, G., Compact finite difference schemes of sixth order for the Helmholtz equation, Journal of Computational and Applied Mathematics, 203, 1, 15-31 (2007) · Zbl 1112.65099 |

[2] | Boisvert, R. F., A fourth-order-accurate Fourier method for the Helmholtz equation in three dimensions, ACM Transactions on Mathematical Software, 13, 3, 221-234 (1987) · Zbl 0626.65104 |

[3] | Tsynkov, S.; Turkel, E., A Cartesian Perfectly Matchedayer for the Helmholtz Equation. Artificial Boundary Conditions with Applications to CEM. Loic Tourvete (2001), Cambridge, Mass, USA: Ovascience, Cambridge, Mass, USA |

[4] | Singer, I.; Turkel, E., High-order finite difference methods for the Helmholtz equation, Computer Methods in Applied Mechanics and Engineering, 163, 1-4, 343-358 (1998) · Zbl 0940.65112 |

[5] | Harari, I.; Hughes, T. J. R., Finite element methods for the Helmholtz equation in an exterior domain: model problems, Computer Methods in Applied Mechanics and Engineering, 87, 1, 59-96 (1991) · Zbl 0760.76047 |

[6] | Mehdizadeh, O. Z.; Paraschivoiu, M., Investigation of a two-dimensional spectral element method for Helmholtz’s equation, Journal of Computational Physics, 189, 1, 111-129 (2003) · Zbl 1024.65112 |

[7] | Nabavi, M.; Siddiqui, M. H. K.; Dargahi, J., A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation, Journal of Sound and Vibration, 307, 3-5, 972-982 (2007) |

[8] | Turkel, E.; Gordon, D.; Gordon, R.; Tsynkov, S., Compact 2D and 3D Sixth order schemes for the Helmholtz equation with variable wave number, Journal of Computational Physics, 232, 1, 272-287 (2013) · Zbl 1291.65273 |

[9] | Gupta, M. M.; Kouatchou, J.; Zhang, J., Comparison of second- and fourth-order discretizations for multigrid Poisson solvers, Journal of Computational Physics, 132, 2, 226-232 (1997) · Zbl 0881.65120 |

[10] | Zhang, J., Fast and high accuracy multigrid solution of the three-dimensional Poisson equation, Journal of Computational Physics, 143, 2, 449-461 (1998) · Zbl 0927.65141 |

[11] | Wesseling, P., An Introduction to Multigrid Methods (1992), Chichester, Uk: John Wiley & Sons, Chichester, Uk · Zbl 0760.65092 |

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