Multigrid method based on transformation-free high-order scheme for solving 2D Helmholtz equation on nonuniform grids
- Fazal Ghaffar^{1},
- Noor Badshah^{2},
- Saeed Islam^{1} and
- Muhammad Altaf Khan^{1}Email author
https://doi.org/10.1186/s13662-016-0745-2
© Ghaffar et al. 2016
Received: 7 June 2015
Accepted: 10 January 2016
Published: 25 January 2016
Abstract
High-order compact difference schemes can achieve higher-order accuracy on uniform grids. However, in some cases these may not achieve the desired accuracy. Therefore, we propose a multigrid method based on high-order compact difference scheme on nonuniform grids. We will use interpolation and restriction operators developed by Ge and Cao (J. Comput. Phys. 230:4051-4070, 2011). The suggested scheme has up to fourth-order accuracy. Lastly, some numerical experiments are given to show the accuracy and performance of the proposed scheme.
Keywords
MSC
1 Introduction
Equation (1) has been solved by different techniques such as finite-difference method (FDM) [4], fast-Fourier-transform-based (FFT) methods [5], finite-element method (FEM) [6], the spectral-element method [7], compact finite-difference method [8], and multigrid methods [9]. The multigrid method based on HOC schemes is among the most efficient iterative techniques for solving PDEs [10, 11].
In FDM the number of mesh points will be enlarged to increase the accuracy; however, it will also increase the computational time. The Helmholtz equation is solved by FEM and spectral-element method, but the limitations of these methods are of high computational cost [7]. Many iterative techniques for the Helmholtz equation suffer due to their slow convergence. The investigation on fast iterative methods to efficiently solve the large algebraic systems arising from high-order difference schemes for PDEs is more attractive. Multigrid methods together with the HOC schemes on uniform mesh sizes are developed in [11–14]. In most cases where sudden changes occur in a flow, the step sizes have to be rectified over the entire domain. Under these situations, where points are concentrated in the regions of sharp variation, local mesh refinement procedures [1, 2, 9, 15–17] are necessary, thus dramatically reducing the computational time and computer storage. Ge and Cao [1, 18] developed a multigrid method with HOC scheme on nonuniform grids for solving 2D convection diffusion equation and 3D Poisson equation. This paper is based on approach that an interpolation operator and a projection operator that are suited for a HOC scheme using nonuniform mesh are represented by a transformation-free HOC scheme on nonuniform grids. The main focus in this paper is to develop a multigrid method based on a HOC scheme on nonuniform grids for solving of the 2D Helmholtz equation. To the best of our knowledge, the 2D Helmholtz equation is not solved by a multigrid method based on a HOC scheme on nonuniform grids.
2 HOC scheme on nonuniform grids
3 Multigrid method
The multigrid method is one of the most efficient and fastest methods for solving PDEs. In the multigrid method, the rate of convergence is independent of the mesh size. This method is more effective for solving large-scale sparse linear systems obtained from the discretization of elliptic PDEs [9, 10, 19–22]. The main principle of the multigrid method is to smoothen the error on coarse grid level using basic iterative methods such as Jacobi or Gauss-Seidel method, etc. The multigrid method consists of three important components that are relaxation, restriction, and interpolation operators. These are applied as ‘a single iteration of a multigrid cycle comprised of manipulating the error by the application of relaxation method, fixing the residuals on the coarse grid level, solving the error equation on the coarse grid and adjusting the correction of coarse grid up to the fine grid level’.
Some specific methods have been applied for the solution of the 2D and 3D Helmholtz equations with HOC schemes on uniform grids [2–4, 6, 8, 12, 13]. A full weighting restriction operator and the standard bilinear interpolation operator are used as the inter-grid transfer operators. But in the case of nonuniform grids, these restriction and interpolation operators cannot be used; so new restriction and interpolation operators for nonuniform grids are proposed by Ge and Cao [1] by using the area law developed by Liu [23]. In the following section, we give out the derivation of the two operators for the completeness.
3.1 Restriction operator
The principle of developing restriction operator is based on the evaluation of the residuals on the coarse grid level with the use of residuals on the fine grid level. In the multigrid method, Liu developed a law for the restriction of the residual [23], known as the area law.
3.2 Interpolation operator
3.3 Relaxation operator (smoother)
In the multigrid method, the relaxation operator is an important operator. Its work is not to remove the errors, but to damp the high-frequency components of the errors on the present grid level. A simple smoother (Gauss-Seidel relaxation) method can efficiently remove the errors in all directions for simple isotropic problems [7, 20], but in case of anisotropic and boundary layer problems, the line Gauss-Seidel [19, 21] and alternating line Gauss-Seidel methods [1, 25–27] are shown to be more robust smoothers. In this paper, we use three relaxations to smooth the residuals on each coarse grid such as the line Gauss-Seidel relaxation, natural Gauss-Seidel relaxation, and Red-black Gauss-Seidel relaxation.
4 Numerical experiments
The number of multigrid V-cycles with two schemes and different values of \(\pmb{k=10,50, 100,500,1\text{,}000}\) and \(\pmb{\lambda=-0.9}\) for Example 1 , where \(\pmb{e^{-5}=10^{-5}}\)
k | N = 16 | N = 32 | N = 64 | N = 128 | |
---|---|---|---|---|---|
CDS | 10 | \(3.4623e^{-5}\) | \(2.4504e^{-5}\) | \(2.1089e^{-5}\) | \(3.2161e^{-5}\) |
50 | \(2.7134e^{-5}\) | \(2.3692e^{-5}\) | \(1.9880e^{-5}\) | \(1.7322e^{-5}\) | |
100 | \(1.7091e^{-5}\) | \(1.6700e^{-5}\) | \(1.4650e^{-5}\) | \(1.3705e^{-5}\) | |
500 | \(1.1898e^{-5}\) | \(1.1662e^{-5}\) | \(1.1497e^{-5}\) | \(1.1403e^{-5}\) | |
1,000 | \(1.1697e^{-5}\) | \(1.3534e^{-5}\) | \(1.9545e^{-5}\) | \(1.6564e^{-5}\) | |
HOC | 10 | \(7.2104e^{-5}\) | \(1.1268e^{-5}\) | \(9.9271e^{-6}\) | \(9.3180e^{-7}\) |
50 | \(5.2183e^{-5}\) | \(1.6971e^{-5}\) | \(9.9128e^{-6}\) | \(9.3345e^{-7}\) | |
100 | \(3.6107e^{-5}\) | \(1.6881e^{-5}\) | \(7.5263e^{-6}\) | \(9.2791e^{-7}\) | |
500 | \(2.1945e^{-5}\) | \(1.5213e^{-5}\) | \(7.6298e^{-6}\) | \(8.4423e^{-7}\) | |
1,000 | \(4.3177e^{-5}\) | \(2.2813e^{-5}\) | \(2.1970e^{-5}\) | \(9.9281e^{-7}\) |
The number of multigrid V-cycles with two schemes and different values of \(\pmb{k=10,50, 100,500,1\text{,}000}\) and \(\pmb{\lambda=0.9}\) for Example 2 , where \(\pmb{e^{-5}=10^{-5}}\)
k | N = 16 | N = 32 | N = 64 | N = 128 | |
---|---|---|---|---|---|
CDS | 10 | \(8.4161e^{-3}\) | \(4.2104e^{-3}\) | \(9.2610e^{-5}\) | \(2.6041e^{-5}\) |
50 | \(8.3413e^{-3}\) | \(4.2390e^{-3}\) | \(9.8707e^{-5}\) | \(2.1374e^{-5}\) | |
100 | \(7.7917e^{-3}\) | \(4.7540e^{-3}\) | \(8.6342e^{-5}\) | \(1.9792e^{-5}\) | |
500 | \(7.6893e^{-3}\) | \(3.6964e^{-3}\) | \(7.4763e^{-5}\) | \(1.4136e^{-5}\) | |
1,000 | \(7.6712e^{-3}\) | \(3.8533e^{-3}\) | \(9.1249e^{-5}\) | \(2.2693e^{-5}\) | |
HOC | 10 | \(3.8844e^{-5}\) | \(2.8600e^{-6}\) | \(8.7210e^{-7}\) | \(5.3421e^{-7}\) |
50 | \(3.2736e^{-5}\) | \(2.9751e^{-6}\) | \(8.1872e^{-6}\) | \(5.3453e^{-7}\) | |
100 | \(3.1673e^{-5}\) | \(2.2360e^{-6}\) | \(8.2453e^{-6}\) | \(5.2198e^{-7}\) | |
500 | \(2.9764e^{-5}\) | \(1.7590e^{-6}\) | \(7.9678e^{-6}\) | \(5.4340e^{-7}\) | |
1,000 | \(4.7337e^{-5}\) | \(3.2187e^{-6}\) | \(7.9711e^{-6}\) | \(5.2891e^{-7}\) |
Example 1
The error norms and order of accuracy of the two schemes for Example 1 , where \(\pmb{e^{-5}=10^{-5}}\) , \(\pmb{\|e\|_{2}}\) , \(\pmb{k = 10}\) , \(\pmb{N = 16,32,64,128}\)
N | λ | \(\boldsymbol{16^{2}} \) | \(\boldsymbol{32^{2}}\) | \(\boldsymbol{64^{2}}\) | \(\boldsymbol{128^{2}}\) | Order |
---|---|---|---|---|---|---|
CDS | 0.0 | \(6.0982e^{-4}\) | \(4.9180e^{-4}\) | \(9.9082e^{-5}\) | \(7.9118e^{-5}\) | 0.310 |
−0.2 | \(4.1044e^{-4}\) | \(3.1322e^{-4}\) | \(7.1160e^{-5}\) | \(5.0032e^{-5}\) | 0.392 | |
−0.4 | \(2.5100e^{-4}\) | \(1.7200e^{-4}\) | \(6.7640e^{-5}\) | \(3.9155e^{-5}\) | 0.545 | |
−0.6 | \(8.1398e^{-5}\) | \(6 .1021e^{-5}\) | \(4.3302e^{-5}\) | \(2.4203e^{-5}\) | 0.415 | |
−0.8 | \(5.8197e^{-5}\) | \(5.0295e^{-5}\) | \(3.0955e^{-5}\) | \(1.5064e^{-5}\) | 0.531 | |
−0.9 | \(3.4623e^{-5}\) | \(2.4504e^{-5}\) | \(2.1089e^{-5}\) | \(3.2161e^{-5}\) | 0.210 | |
HOC | 0.0 | \(4.8122e^{-4}\) | \(3.6113e^{-4}\) | \(9.4102e^{-5}\) | \(5.8410e^{-5}\) | 0.414 |
−0.2 | \(1.1438e^{-4}\) | \(1.0661e^{-4}\) | \(6.3918e^{-5}\) | \(4.1235e^{-5}\) | 0.101 | |
−0.4 | \(9.1100e^{-5}\) | \(6.1818e^{-5}\) | \(3.2561e^{-5}\) | \(1.9760e^{-5}\) | 0.559 | |
−0.6 | \(7.1398e^{-5}\) | \(4.5502e^{-5}\) | \(1.6021e^{-5}\) | \(6.2030e^{-6}\) | 0.650 | |
−0.8 | \(4.7341e^{-5}\) | \(8.8061e^{-6}\) | \(7.6660e^{-6}\) | \(3.7210e^{-7}\) | 0.242 | |
−0.9 | \(7.2104e^{-5}\) | \(1.1268e^{-5}\) | \(9.9271e^{-6}\) | \(9.3180e^{-7}\) | 2.677 |
The number of multigrid V-cycles and CPU time with two schemes and different relaxation methods with \(\pmb{32^{2}}\) for Example 1
Scheme | λ | Line GS | Red-black GS | Natural GS | |||
---|---|---|---|---|---|---|---|
Iterations | CPU | Iterations | CPU | Iterations | CPU | ||
CDS | 0.0 | 9 | 0.334 | 10 | 0.350 | 12 | 1.840 |
−0.2 | 10 | 0.362 | 11 | 0.432 | 13 | 1.920 | |
−0.4 | 10 | 1.400 | 12 | 1.860 | 15 | 2.940 | |
−0.6 | 11 | 1.521 | 12 | 2.032 | 17 | 3.660 | |
−0.8 | 21 | 2.230 | 36 | 4.564 | 43 | 6.583 | |
−0.9 | 33 | 3.345 | 43 | 7.216 | 55 | 8.000 | |
HOC | 0.0 | 9 | 0.300 | 9 | 0.810 | 12 | 1.832 |
−0.2 | 10 | 0.340 | 11 | 0.830 | 12 | 1.980 | |
−0.4 | 11 | 0.810 | 12 | 1.767 | 14 | 2.200 | |
−0.6 | 11 | 0.900 | 12 | 2.203 | 18 | 3.80 | |
−0.8 | 12 | 1.800 | 13 | 3.720 | 22 | 4.960 | |
−0.9 | 15 | 2.200 | 21 | 6.318 | 33 | 6.550 |
Example 2
The error norms and order of accuracy of the two schemes for Example 2 , where \(\pmb{e^{-5}=10^{-5}}\) , \(\pmb{\|e\|_{2}}\) , \(\pmb{k = 10}\) , \(\pmb{N = 16,32,64,128 }\)
N | λ | \(\boldsymbol{16^{2}} \) | \(\boldsymbol{32^{2}}\) | \(\boldsymbol{64^{2}}\) | \(\boldsymbol{128^{2}}\) | Order |
---|---|---|---|---|---|---|
CDS | 0.0 | \(5.2002e^{-2}\) | \(4.8890e^{-2}\) | \(3.2290e^{-2}\) | \(7.1098e^{-3}\) | 0.089 |
0.2 | \(3.4421e^{-2}\) | \(2.2213e^{-2}\) | \(1.1180e^{-2}\) | \(5.3112e^{-3}\) | 0.631 | |
0.4 | \(2.5541e^{-2}\) | \(1.7002e^{-2}\) | \(1.1040e^{-2}\) | \(3.1255e^{-4}\) | 0.587 | |
0.6 | \(1.1983e^{-2}\) | \(7.2100e^{-3}\) | \(4.1033e^{-3}\) | \(6.1043e^{-5}\) | 0.732 | |
0.8 | \(5.1287e^{-3}\) | \(1.9520e^{-3}\) | \(7.1530e^{-5}\) | \(1.6054e^{-5}\) | 0.999 | |
0.9 | \(8.4161e^{-3}\) | \(4.2104e^{-3}\) | \(9.2610e^{-5}\) | \(2.6041e^{-5}\) | 1.393 | |
HOC | 0.0 | \(4.2122e^{-2}\) | \(3.3211e^{-3}\) | \(7.4412e^{-4}\) | \(5.8234e^{-4}\) | 3.664 |
0.2 | \(1.1218e^{-2}\) | \(1.6601e^{-3}\) | \(6.3318e^{-4}\) | \(4.7715e^{-5}\) | 2.756 | |
0.4 | \(9.7110e^{-3}\) | \(6.1218e^{-4}\) | \(3.6152e^{-5}\) | \(3.1961e^{-6}\) | 3.987 | |
0.6 | \(3.9328e^{-4}\) | \(4.1155e^{-5}\) | \(2.6001e^{-6}\) | \(2.3087e^{-6}\) | 3.256 | |
0.8 | \(3.7022e^{-5}\) | \(1.8676e^{-6}\) | \(5.2260e^{-7}\) | \(3.3350e^{-7}\) | 4.309 | |
0.9 | \(3.8844e^{-5}\) | \(2.8600e^{-6}\) | \(8.7210e^{-7}\) | \(5.3421e^{-7}\) | 3.763 |
The number of multigrid V-cycles and CPU time with two schemes and different relaxation methods with \(\pmb{32^{2}}\) for Example 2
Scheme | λ | Line GS | Red-black GS | Natural GS | |||
---|---|---|---|---|---|---|---|
Iterations | CPU | Iterations | CPU | Iterations | CPU | ||
CDS | 0.0 | 11 | 0.534 | 12 | 0.550 | 12 | 1.480 |
−0.2 | 11 | 0.562 | 12 | 0.630 | 13 | 1.960 | |
−0.4 | 11 | 0.584 | 12 | 0.980 | 17 | 2.488 | |
−0.6 | 12 | 1.765 | 13 | 1.382 | 19 | 3.600 | |
−0.8 | 17 | 1.936 | 19 | 1.996 | 33 | 5.853 | |
−0.9 | 27 | 3.145 | 33 | 4.162 | 43 | 6.980 | |
HOC | 0.0 | 9 | 0.310 | 9 | 0.532 | 12 | 1.330 |
−0.2 | 9 | 0.330 | 10 | 0.572 | 12 | 1.860 | |
−0.4 | 9 | 0.415 | 11 | 0.677 | 13 | 2.200 | |
−0.6 | 10 | 0.970 | 12 | 1.238 | 18 | 2.890 | |
−0.8 | 11 | 1.770 | 16 | 1.720 | 22 | 3.973 | |
−0.9 | 13 | 2.245 | 19 | 4.080 | 23 | 5.560 |
The number of multigrid V-cycles and CPU time with two schemes and different relaxation methods with \(\pmb{32^{2}}\) for Example 2
Scheme | Grids | Line GS | Red-black GS | Natural GS | |||
---|---|---|---|---|---|---|---|
Iterations | CPU | Iterations | CPU | Iterations | CPU | ||
CDS | \( 8^{2} \) | 9 | 0.060 | 11 | 0.140 | 11 | 0.800 |
\(16^{2} \) | 11 | 0.062 | 12 | 0.840 | 11 | 1.392 | |
\(32^{2}\) | 11 | 0.284 | 12 | 1.220 | 12 | 2.268 | |
\(64^{2}\) | 12 | 2.652 | 13 | 2.842 | 13 | 3.230 | |
\(128^{2}\) | 13 | 2.920 | 15 | 3.296 | 15 | 3.838 | |
HOC | \( 8^{2} \) | 8 | 0.070 | 8 | 0.532 | 12 | 0.720 |
\(16^{2} \) | 8 | 0.073 | 8 | 0.660 | 12 | 0.960 | |
\(32^{2}\) | 9 | 0.210 | 9 | 0.977 | 12 | 1.862 | |
\(64^{2} \) | 9 | 2.510 | 10 | 1.322 | 13 | 2.890 | |
\(128^{2} \) | 11 | 2.872 | 12 | 2.260 | 18 | 3.713 |
Example 3
The error norms and CPU (seconds) for a multigrid method with different discretized schemes for Example 3 , \(\pmb{\|e\|_{2}}\) , \(\pmb{k = 10}\)
N | CDS scheme | CPU (seconds) | Order | HOC scheme | Order |
---|---|---|---|---|---|
4 | \(7.5620e^{-2}\) | 0.060 | 2.01 | \(6.3220e^{-4}\) | 2.40 |
8 | \(3.2228e^{-3}\) | 0.074 | 4.55 | \(8.3311e^{-5}\) | 2.92 |
16 | \(4.0644e^{-4}\) | 0.088 | 2.98 | \(3.2034e^{-6}\) | 4.70 |
32 | \(8.1510e^{-5}\) | 1.03 | 2.31 | \(8.1284e^{-7}\) | 1.97 |
64 | \(2.0389e^{-6}\) | 1.80 | 5.32 | \(7.3819e^{-8}\) | 3.46 |
128 | \(5.5090e^{-7}\) | 2.90 | 1.43 | \(6.1901e^{-8}\) | 0.25 |
5 Conclusion
In this paper, we have proposed a transformation-free high-order compact finite-difference scheme on nonuniform grids for solution of the 2D Helmholtz equation to get up to fourth-order accuracy. Furthermore, we have applied the multigrid method based on the HOC scheme on nonuniform grids, which solved the resulting system efficiently. In the case of boundary layer problems with suitable grid stretching ratios, the accuracy is up to fourth order for the HOC scheme and second order for the CD scheme. Numerical results show that the multigrid method with HOC scheme has the required accuracy and is faster than the CD scheme.
Declarations
Acknowledgements
The authors would like to express their gratitude to the editors and anonymous reviewers for their valuable suggestions, which substantially improved the standard of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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