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An extrapolation full multigrid algorithm combined with fourth-order compact scheme for convection–diffusion equations
- Ming Li^{1, 2},
- Zhoushun Zheng^{1} and
- Kejia Pan^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-018-1631-x
© The Author(s) 2018
Received: 11 January 2018
Accepted: 3 May 2018
Published: 10 May 2018
Abstract
In this paper, we propose an extrapolation full multigrid (EXFMG) algorithm to solve the large linear system arising from a fourth-order compact difference discretization of two-dimensional (2D) convection diffusion equations. A bi-quartic Lagrange interpolation for the solution on previous coarser grid is used to construct a good initial guess on the next finer grid for V- or W-cycles multigrid solver, which greatly reduces the number of relaxation sweeps. Instead of performing a fixed number of multigrid cycles as used in classical full multigrid methods, a series of grid level dependent relative residual tolerances is introduced to control the number of the multigrid cycles. Once the fourth-order accurate numerical solutions are obtained, a simple method based on the midpoint extrapolation is employed for the fourth-order difference solutions on two-level grids to construct a sixth-order accurate solution on the entire fine grid cheaply and directly. Numerical experiments are conducted to verify that the proposed method has much better efficiency compared to classical multigrid methods. The proposed EXFMG method can also be extended to solve other kinds of partial differential equations.
Keywords
- Extrapolation
- Full multigrid method
- Compact difference scheme
- Convection–diffusion equation
MSC
- 65N06
- 65N55
1 Introduction
Richardson extrapolation [1, 2] is an acceleration method, used to improve the rate of convergence for a sequence. In 1983, Marchuk and Shaidurov extended Richardson extrapolation technique in the framework of the finite difference (FD) method [3]. Later on, Blum et al. discussed Richardson extrapolation strategy in conjunction with the finite element (FE) method [4–7].
Multigrid methods (MGs) [8, 9] have been shown to be one of the most efficient modern numerical strategies to solve the large linear systems arising from FE or FD discretizations of partial differential equations (PDEs), such as the Poisson equation [10–14], Helmholtz equation [15–17], convection–diffusion equation [18–20]. The main advantage of the multigrid method is that its convergence rate is independent of the number of unknowns [8, 21]. The full multigrid (FMG) method, which combines nested iteration with a V- or W-cycles multigrid method, is well known to be optimal with respect to the energy norm and the \(L^{2}\)-norm [8, 21–23]. It starts on the coarsest grid with a direct solver, then the solution is interpolated to form the initial guess on the next finer grid, where a few V- or W-cycles are used to produce an approximate solution whose algebraic error matches with the discretization error. The procedure goes on recursively until the finest grid.
Recently, the Richardson extrapolation technique has been applied in multigrid methods. One group of researchers used Richardson extrapolation to construct a good initial guess for the iterative solver. An extrapolation cascadic multigrid (EXCMG) method [24, 25] was proposed by Chen et al. to solve second-order elliptic problems. This method uses a new extrapolation formula to construct a quite good initial guess for the iterative solution on the next finer grid, which greatly improves the convergence rate of the original CMG algorithm (see [24–26] for details). Then the EXCMG method has been successfully applied to non-smooth elliptic problems [27, 28], parabolic problems [29], and some other related problems [30–32]. Moreover, Pan [33] and Li [34, 35] developed some EXCMG methods combined with high-order compact difference schemes to solve Poisson equations. In 2017, Hu et al. [36] obtained the super-optimality of the EXCMG method under the energy norm for \(H^{2+\alpha }\)-regular \((0<\alpha \leq 1)\) elliptic problems.
Another group of researchers made efforts to design an iterative procedure with Richardson extrapolation strategy to enhance the solution accuracy on the finest grid. Sun, Zhang and Dai developed a sixth-order FD scheme for solving the 2D convection–diffusion equation [37, 38]. In their approach, the ADI method is applied to compute the fourth-order accurate solution on the fine and coarse grids, respectively, then the Richardson extrapolation technique and an operator-based interpolation scheme are employed in each ADI iteration to calculate the sixth-order accurate solution on the fine grid. In 2009, Wang and Zhang designed an explicit sixth-order compact discretization strategy (MG-Six) for the 2D Poisson equation [39]. They used a V-cycle multigrid method to get the fourth-order accurate solutions on both the fine and the coarse grids first, and then chose the iterative operator with Richardson extrapolation technique to compute the sixth-order accurate solution on the fine grid.
We extend the idea described in the literature [32, 33, 39] to the original full multigrid method, and develop an extrapolation full multigrid (EXFMG) method to solve the 2D convection–diffusion equation with a fourth-order compact difference discretization. To be more precise, a bi-quartic Lagrange interpolation operator on a coarser grid is applied to get a good initial guess on the next finer grid for the multigrid V- or W-cycles solver. Besides, a stopping criterion related to relative residual is used to conveniently obtain the numerical solution with the desired accuracy. Moreover, a mid-point extrapolation strategy is used to obtain cheaply and directly a sixth-order accurate solution on the entire fine grid from two fourth-order solutions on two different grids (current fine and previous coarse grids). Finally, through two examples chosen from the literature, the computational efficiency of the EXFMG method is discussed in detail.
The rest of the paper is organized as follows. Section 2 introduces the model problem and a fourth-order compact difference scheme. Section 3 depicts the classical multigrid method. And the extrapolation full multigrid method is described in Sect. 4. Numerical results are given in Sect. 5 to show the efficiency and high-order accuracy of the new algorithm. Finally, concluding remarks are provided in Sect. 6.
2 Model problem and discretization
Let \(\Omega =[0,M_{x}]\times [0,M_{y}] \) and divide Ω into \(N_{x} \times N_{y}\) uniform grids with the mesh size \(\Delta x = \frac{ {M_{x} }}{{N_{x} }}\) and \(\Delta y = \frac{{M_{y} }}{{N_{y} }}\) in the x and y coordinate directions, respectively. \(N_{x}\) and \(N_{y}\) are the numbers of uniform intervals in the x and y directions, respectively. The mesh points are \((x_{i},y _{j})\) with \(x_{i}=i\Delta x\) and \(y_{i}=i\Delta y\), \(0 \le i \le N _{x}\), \(0 \le j \le N_{y}\). For simplicity, we use \(U_{i,j}\) to represent the approximation of \(u(x_{i},y_{j})\) at the mesh point \((x_{i},y_{j}) \), set \(f_{i,j}=f(x_{i},y_{j})\), \(p_{i,j}=p(x_{i},y_{j})\), and so on.
3 Classical multigrid method
4 Extrapolation full multigrid method
4.1 Description of the EXFMG algorithm
- (1)
A bi-quartic Lagrange interpolation operator on coarse grid is taken as a full interpolation to provide a good initial guess on the next finer grid for multigrid iteration.
- (2)
Instead of performing a fixed number of multigrid iterations at each level of grid as used in the usual FMG method, a tolerance \(\epsilon _{j}\) related to the relative residual is used on jth grid level (see line 4 in Algorithm 3), which enables us to conveniently calculate the approximate solution with the desired accuracy and keep suitable computational cost.
- (3)
The FMG method only computes the fourth-order accuracy solution of each discretization system. But the extrapolation method based on the midpoint formula (see line 5 in Algorithm 3 and Sect. 4.2) enhances the solution accuracy on each grid (see numerical results section for details).
4.2 Fine mesh extrapolation operator
In the classical Richardson extrapolation formula, two solutions on fine and coarse grids are used to obtain an extrapolation solution on the coarse grid. Based on the Richardson extrapolation technique and interpolation theory, in 2017, Pan proposed a mid-point extrapolation technique to enhance the accuracy of approximations on fine grid directly and cheaply (see [33] for details). Now, we will introduce this strategy for 2D case in the following.
Remark 4.1
It should be noted that the extrapolation interpolator \(FMRE\) is a local operation, which can be done very effectively.
4.3 Computational work
Throughout this paper, when using Algorithm 1 as a solver on each grid of the EXFMG method, we always set \(N_{\textrm{cycles}} =1\). Now we turn to estimate the computational cost of the EXFMG method in terms of \(work\) \(units\) (WUs). Roughly speaking, “1 WU” is the computational cost of performing one relaxation sweep on the finest grid. For simplicity, we neglect the amount of work needed to solve the equations on the coarsest grid and the transfer operators (include interpolation and restriction) between different levels of grids since it normally counts 10–20% of the total cost of the algorithm.
5 Numerical experiments
In this section, we conduct numerical experiments with two examples both on the unit square domain \(\Omega =(0,1)\times (0,1)\). In the two test problems, the right-hand side function \(f(x,y)\) and the Dirichlet boundary conditions on ∂Ω are prescribed to satisfy the given analytic solutions \(u(x,y)\). Our codes are written in Matlab and the programs are carried out on a desktop with Inter (R) Core(TM) i5-6200 U CPU (2.30 GHZ, 2.40 GHZ) and 4 GB RAM.
For comparison purposes, we denote the Algorithm 1 with \(N_{\textrm{cycles}} =1\) as MG. The iterative procedure of MG method starts with zero initial guesses and terminates when the approximation solution \(u_{L}^{*}\) on the finest grid satisfies the same stopping criterion \({{\Vert F_{L} - A_{L} u_{L}^{*} \Vert }} \le \epsilon _{L} {{\Vert F_{L} \Vert }}\) as the EXFMG algorithm.
In MG and EXFMG methods with L embedded grids, the standard bilinear interpolation is used to transfer corrections from the coarse grid to the fine grid, the full-weighting scheme is employed to project residual from the fine grid to the coarse grid, the Conjugate Gradient (CG) method is chosen as a smoother to eliminate the high frequency error, and the number of pre-smoothing and post-smoothing steps are set to be \(v_{1}=v_{2}=2\). \(\epsilon_{L}=10^{-10}\) is taken as the stopping tolerance on the finest grid.
Example 5.1
Example 5.2
([40])
Errors and costs of algorithms with \(L=5\) for different discretization system arising from Example 5.1
\({\Omega _{L}} \) | Method | \(\Vert u-u_{L}^{*} \Vert _{\infty }\) | \(\Vert u-u_{L}^{*} \Vert _{2}\) | CPU | Iter | WU | Tflops |
---|---|---|---|---|---|---|---|
256 × 256 | MG | 6.30E−06 | 2.41E−06 | 0.92 | 9 | 47.8 | 1.5 |
EXFMG | 1.23E−07 | 2.17E−08 | 0.91 | 6, 8, 10, 1 | 45.6 | 1.1 | |
512 × 512 | MG | 3.94E−07 | 1.51E−07 | 3.42 | 9 | 47.8 | 23.5 |
EXFMG | 2.05E−09 | 3.41E−10 | 3.05 | 6, 7, 9, 1 | 43.9 | 16.8 | |
1024 × 1024 | MG | 2.47E−08 | 9.45E−09 | 14.3 | 9 | 47.8 | 377.9 |
EXFMG | 7.10E−11 | 6.33E−12 | 12.1 | 5, 7, 8, 1 | 38.3 | 229.3 | |
2048 × 2048 | MG | 1.54E−09 | 5.91E−10 | 43.4 | 10 | 53.1 | 6736.5 |
EXFMG | 7.34E−12 | 8.68E−13 | 40.5 | 5, 7, 8, 1 | 38.3 | 3679.8 |
Errors and costs of algorithms with \(L=5\) for different discretization system arising from Example 5.2
\({\Omega _{L}} \) | Method | \(\Vert u-u_{L}^{*} \Vert _{\infty }\) | \(\Vert u-u_{L}^{*} \Vert _{2}\) | CPU | Iter | WU | Tflops |
---|---|---|---|---|---|---|---|
256 × 256 | MG | 3.15E−05 | 3.46E−06 | 0.92 | 8 | 42.5 | 1.3 |
EXFMG | 3.02E−06 | 2.84E−07 | 1.21 | 7, 9, 11, 1 | 52.5 | 1.2 | |
512 × 512 | MG | 1.94E−06 | 2.13E−07 | 4.34 | 9 | 47.8 | 23.5 |
EXFMG | 6.18E−08 | 4.40E−09 | 3.95 | 5, 8, 10, 1 | 40.3 | 14.4 | |
1024 × 1024 | MG | 1.21E−07 | 1.33E−08 | 14.95 | 8 | 42.5 | 335.9 |
EXFMG | 1.78E−09 | 3.38E−10 | 12.47 | 4, 6, 9 , 1 | 32.0 | 184.8 | |
2048 × 2048 | MG | 7.53E−09 | 8.35E−10 | 40.64 | 8 | 42.5 | 5389.2 |
EXFMG | 3.65E−10 | 8.29E−11 | 41.84 | 4, 5, 7, 1 | 30.1 | 2920.2 |
Comparison of the \(L^{\infty }\) errors and convergence order on different grid level of MG and EXFMG methods for Example 5.1
j | \(\Omega _{j}\) | MG | EXFMG | ||||
---|---|---|---|---|---|---|---|
\(\Vert u-u_{j}^{*} \Vert _{\infty }\) | Order | \(\Vert u-\bar{u}^{*}_{j} \Vert _{\infty }\) | Order | \(\Vert u-u_{j}^{*} \Vert _{\infty }\) | Order | ||
2 | 64 × 64 | 1.56E−03 | – | 1.56E−03 | – | 3.44E−04 | – |
3 | 128 × 128 | 1.00E−04 | 3.96 | 1.00E−04 | 3.96 | 6.96E−06 | 5.63 |
4 | 256 × 256 | 6.30E−06 | 3.99 | 6.30E−06 | 3.99 | 1.23E−07 | 5.83 |
5 | 512 × 512 | 3.94E−07 | 4.00 | 3.94E−07 | 4.00 | 2.04E−09 | 5.91 |
6 | 1024 × 1024 | 2.47E−08 | 4.00 | 2.47E−08 | 4.00 | 7.10E−11 | 4.84 |
For the accuracy, we find that the numerical solutions from the EXFMG method are much more accurate than those from the MG method (see Tables 1 and 2, Figs. 2 and 3). This means that, to achieve the same accuracy, fewer grid points are needed for the EXFMG method than for the MG method. For instance, the numerical solution by the EXFMG method on the \(256\times 256\) grid is comparable to the numerical solution obtained by the MG method on the \(512\times 512\) grid in Tables 1 and 2.
Comparison of the \(L^{\infty }\) errors and convergence order on different grid level of MG and EXFMG methods for Example 5.2
j | \(\Omega _{j}\) | MG | EXFMG | ||||
---|---|---|---|---|---|---|---|
\(\Vert u-u_{j}^{*} \Vert _{\infty }\) | Order | \(\Vert u-\bar{u}^{*}_{j} \Vert _{\infty }\) | Order | \(\Vert u-u_{j}^{*} \Vert _{\infty }\) | Order | ||
2 | 64 × 64 | 1.16E−02 | – | 1.16E−02 | – | 4.14E−01 | – |
3 | 128 × 128 | 5.36E−04 | 4.43 | 5.36E−04 | 4.43 | 2.01E−04 | 11.01 |
4 | 256 × 256 | 3.15E−05 | 4.09 | 3.15E−05 | 4.09 | 3.02E−06 | 6.06 |
5 | 512 × 512 | 1.94E−06 | 4.02 | 1.94E−06 | 4.02 | 6.18E−08 | 5.61 |
6 | 1024 × 1024 | 1.21E−07 | 4.01 | 1.21E−07 | 4.01 | 1.78E−09 | 5.12 |
Comparison of the \(L^{2}\) errors and convergence order on different grid level of MG and EXFMG methods for Example 5.1
j | \(\Omega _{j}\) | MG | EXFMG | ||||
---|---|---|---|---|---|---|---|
\(\Vert u-u_{j}^{*} \Vert _{\infty }\) | Order | \(\Vert u-\bar{u}^{*}_{j} \Vert _{\infty }\) | Order | \(\Vert u-u_{j}^{*} \Vert _{\infty }\) | Order | ||
2 | 64 × 64 | 5.97E−04 | – | 5.97E−04 | – | 7.81E−05 | – |
3 | 128 × 128 | 3.84E−05 | 3.96 | 3.84E−05 | 3.96 | 1.35E−06 | 5.85 |
4 | 256 × 256 | 2.41E−06 | 3.99 | 2.41E−06 | 3.99 | 2.17E−08 | 5.96 |
5 | 512 × 512 | 1.51E−07 | 4.00 | 1.51E−07 | 4.00 | 3.41E−10 | 5.99 |
6 | 1024 × 1024 | 9.45E−09 | 4.00 | 9.45E−09 | 4.00 | 6.27E−12 | 5.77 |
Comparison of the \(L^{2}\) errors and convergence order on different grid level of MG and EXFMG methods for Example 5.2
j | \(\Omega _{j}\) | MG | EXFMG | ||||
---|---|---|---|---|---|---|---|
\(\Vert u-u_{j}^{*} \Vert _{\infty }\) | Order | \(\Vert u-\bar{u}^{*}_{j} \Vert _{\infty }\) | Order | \(\Vert u-u_{j}^{*} \Vert _{\infty }\) | Order | ||
2 | 64 × 64 | 1.37E−03 | – | 1.37E−03 | – | 1.59E−01 | – |
3 | 128 × 128 | 5.85E−05 | 4.55 | 5.85E−05 | 4.55 | 2.23E−05 | 12.80 |
4 | 256 × 256 | 3.46E−06 | 4.08 | 3.46E−06 | 4.08 | 2.84E−07 | 6.29 |
5 | 512 × 512 | 2.13E−07 | 4.02 | 2.13E−07 | 4.02 | 4.40E−09 | 6.01 |
6 | 1024 × 1024 | 1.33E−08 | 4.00 | 1.33E−08 | 4.00 | 3.38E−10 | 3.70 |
Comparison of the iteration number and computational cost (WU) on different grid level of MG and EXFMG methods for Example 5.1
j | \(\Omega _{j}\) | MG | EXFMG | ||
---|---|---|---|---|---|
Iter | WU | Iter | WU | ||
2 | 64 × 64 | 1 | 0.02 | 1 | 0.02 |
3 | 128 × 128 | 9 | 0.70 | 10 | 0.78 |
4 | 256 × 256 | 9 | 2.95 | 8 | 2.63 |
5 | 512 × 512 | 9 | 11.95 | 7 | 9.30 |
6 | 1024 × 1024 | 9 | 47.95 | 5 | 26.64 |
Comparison of the iteration number and computational cost (WU) on different grid level of MG and EXFMG methods for Example 5.2
j | \(\Omega _{j}\) | MG | EXFMG | ||
---|---|---|---|---|---|
Iter | WU | Iter | WU | ||
2 | 64 × 64 | 1 | 0.02 | 1 | 0.02 |
3 | 128 × 128 | 9 | 0.70 | 11 | 0.86 |
4 | 256 × 256 | 8 | 2.63 | 9 | 2.95 |
5 | 512 × 512 | 9 | 11.95 | 6 | 7.97 |
6 | 1024 × 1024 | 8 | 42.63 | 4 | 21.31 |
From the above discussion, we can see that the EXFMG method not only achieves high-order accuracy but also keeps a low computational cost. Hence, it is a cost-effective numerical solver.
6 Conclusion
We present an EXFMG method combined with a fourth-order compact difference scheme, a bi-quartic Lagrange interpolation and a simple extrapolation technique, to solve the 2D convection–diffusion equation. Numerical experiments were conducted for two test problems to demonstrate that the proposed EXFMG method improves the solution accuracy and keeps less computational costs, compared to the classical MG method.
We need point out that the fourth-order compact difference method, which is employed to discretize the differential equation, can be replaced by other numerical methods, such as the FE method and the finite volume method. Furthermore, we also plan to extend this study and apply EXFMG algorithm with sixth-order combined compact difference scheme [43–45] to solve unsteady three-dimensional convection–diffusion equations.
Declarations
Acknowledgements
The research was supported by the National Natural Science Foundation of China (Nos. 41474103, 11661027, 11601012), the National Key Research and Development Program of China (Nos. 2017YFB0701700, 2017YFB0305601), the Excellent Youth Foundation of Hunan Province of China (No. 2018JJ1042), the Natural Science Foundation of Yunnan Province of China (No. 2017FH001-012), the Innovation-Driven Project of Central South University (No. 2018CX042) and the Reserve Talents Foundation of Honghe University (No. 2015HB0304). We are grateful to the two anonymous reviewers for their helpful comments.
Authors’ contributions
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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