- Research
- Open Access
An efficient parallel algorithm for Caputo fractional reaction-diffusion equation with implicit finite-difference method
- Qinglin Wang^{1, 2}Email author,
- Jie Liu^{1, 2},
- Chunye Gong^{1, 2},
- Xiantuo Tang^{1, 2},
- Guitao Fu^{3} and
- Zuocheng Xing^{1, 2}
https://doi.org/10.1186/s13662-016-0929-9
© Wang et al. 2016
- Received: 7 October 2015
- Accepted: 28 July 2016
- Published: 11 August 2016
Abstract
An efficient parallel algorithm for Caputo fractional reaction-diffusion equation with implicit finite-difference method is proposed in this paper. The parallel algorithm consists of a parallel solver for linear tridiagonal equations and parallel vector arithmetic operations. For the parallel solver, in order to solve the linear tridiagonal equations efficiently, a new tridiagonal reduced system is developed with an elimination method. The experimental results show that the parallel algorithm is in good agreement with the analytic solution. The parallel implementation with 16 parallel processes on two eight-core Intel Xeon E5-2670 CPUs is 14.55 times faster than the serial one on single Xeon E5-2670 core.
Keywords
- fractional reaction-diffusion equation
- parallel computing
- elimination method
- tridiagonal reduced system
MSC
- 34A08
- 65Y05
1 Introduction
Fractional differential equations (FDEs) refer to a class of differential equations which use derivatives of non-integer order [1, 2]. Fractional equations have proved to be very reliable models for many scientific and engineering problems [3, 4]. Because it is difficult to solve complex fractional problems analytically, more and more work focuses on numerical solutions [5, 6].
Recently, there has been great interest in FDEs [7–12]. Ahmad et al. [7] discussed the existence of the solution of a Caputo fractional reaction-diffusion equation with various boundary conditions. Rida et al. [8] applied the generalized differential transform method to solve nonlinear fractional reaction-diffusion partial differential equations. Chen et al. used the explicit finite-difference approximation [9] and implicit difference approximation [10] to solve the Riesz space fractional reaction-dispersion equation.
The numerical methods for FDEs include finite-difference methods [13, 14], finite element methods [15] and spectral methods [16–20]. Fractional reaction-diffusion equations are related to spatial and time coordinates, so the numerical solutions are often time-consuming. Large scale applications and algorithms in science and engineering such as neutron transport [21–23], computational fluid dynamics [24–26], large sparse systems [27] rely on parallel computing [24, 28, 29]. In order to overcome the difficulty, parallel computing has been introduced into the numerical solutions for fractional equations [30–32]. Kai Diethelm [31] parallelized the fractional Adams-Bashforth-Moulton algorithm for the first time and the execution time of the algorithm was efficiently reduced. Gong et al. [33] presented a parallel solution for time fractional reaction-diffusion equation with explicit difference method. In the parallel solver, the technology named pre-computing fractional operator was used to optimize performance.
In this paper, we address an efficient parallel algorithm for time fractional reaction-diffusion equation with an implicit finite-difference method. In this parallel algorithm, the system of linear tridiagonal equations, vector-vector additions and constant-vector multiplications are efficiently processed in parallel. The linear tridiagonal system is parallelized with a new elimination method, which is effective and has simplicity in computer programming. The results indicate there is no significant difference between the implementation and the exact solution. The parallel algorithm with 16 parallel processes on two eight-core Intel Xeon E5-2670 CPUs runs 14.55 times faster than the serial algorithm.
2 Background
2.1 Numerical solution with implicit finite difference
3 Parallel algorithm
3.1 Analysis
In order to get \(U^{n}\) from equation (8), two steps need to be processed. One step performs the right-sided computation of equation (8). Set \(V^{n}=\sum_{k=1}^{n-1}r_{n-1-k}U^{k} + b_{n-1}U^{0} + \sigma F^{n}\). The calculation of \(V^{n}\) mainly involves the constant-vector multiplications and vector-vector additions. The other step solves the tridiagonal linear equations \(AU^{n}=V^{n}\). The constant-vector multiplications and vector-vector additions can easily be parallelized. So we only analyze the parallel implementation of solving the tridiagonal linear equations.
For strictly diagonally dominant linear systems, one parallel approximation algorithm has been proposed [34]. For given α and N, the dominance factor would be very close to one with big M. The approximation algorithm would decrease the precision of the solution and even exacerbate the convergence. In other words, applying the algorithm to this solution of tridiagonal linear equations with fixed M will obviously increase the number of time steps in order to keep the same precision. Thus, it is necessary to solve tridiagonal linear equations accurately. Moreover, each iteration on time step involves one system of tridiagonal linear equations. The right-hand side \(V^{n}\) varies while the tridiagonal matrix A keeps constant in all time iterations. In order to avoid the repeated calculations, the transformation of the tridiagonal matrix is recorded in the parallel solution of tridiagonal linear equations.
3.2 Parallel solution of tridiagonal linear equations
Line 3 allocates \(D_{i}\), \(g_{(i-1)k+1}\), \(c_{jk}\), \(V_{i}\) to the ith process, \(i=1, \ldots, p\).
We can find the coefficient matrix in equation (19) is still a tridiagonal matrix. Line 13 means the reduced system of equations is solved only with processes \(P_{1}\) and \(P_{p}\). \(P_{1}\) and \(P_{p}\) deal with the reduced system of equations using the functions in lines 9-11 as well. \(P_{1}\) and \(P_{p}\) exchange the qkth and \(qk+1\)th equations after elimination, and both can obtain \(u_{(q+1)k}\) and \(u_{(q+1)k+1}\). In process \(P_{1}\), \(u_{(i-1)k+1}\) and \(u_{ik}\) are acquired and sent to \(P_{i}\) (\(1 < i \le q\)). In process \(P_{p}\), \(u_{(i-1)k+1}\) and \(u_{ik}\) are got and dispatched to \(P_{i}\) (\(q+1 \le i < p\)). After \(P_{i}\) (\(1 < i < p\)) receives \(u_{(i-1)k+1}\) and \(u_{ik}\) from \(P_{1}\) or \(P_{p}\), \(U_{i}\) can be solved using equation (17). \(U_{1}\) and \(U_{p}\) can also be figured out by equation (16) and equation (18), respectively.
3.3 Implementation
4 Experimental results and discussion
The specifications of the experiment’s platform
CPU | 2 Intel Xeon E5-2670 CPUs, 8 cores/CPU, 2.6 GHz |
Host OS | Linux Red Hat 4.4.5-6 |
Compiler Version | Intel v13.0.0 |
MPI Version | Intel v4.0.3 |
4.1 Accuracy of parallel solution
Result comparison between the exact analytic solution and the parallel solution at time \(\pmb{t = 1.0}\) and \(\pmb{p=8}\) , where different M and N are applied
M = N | 65 | 129 | 257 | 513 | 1,025 | 2,049 |
---|---|---|---|---|---|---|
Max error | 7.47E − 04 | 3.07E − 04 | 1.26E − 04 | 5.12E − 05 | 2.08E − 05 | 8.47E − 06 |
4.2 Performance improvement
Performance comparison between serial solution (SS) and parallel solution (PS) when different M and N are applied
M = N | 65 | 129 | 257 | 513 | 1,025 | 2,049 | |
---|---|---|---|---|---|---|---|
Runtime (s) | SS | 5.20E − 04 | 3.31E − 03 | 1.22E − 02 | 7.35E − 02 | 4.93E − 01 | 4.03E + 00 |
PS | 6.51E − 04 | 1.42E − 03 | 3.75E − 03 | 1.15E − 02 | 4.56E − 02 | 2.77E − 01 | |
Speedup | 0.80 | 2.34 | 3.26 | 6.37 | 10.80 | 14.55 |
4.3 Scalability
Performance of parallel solution with \(\pmb{M = N = 2\text{,}049}\) when different numbers of processes are applied
No. of processes | 1 | 2 | 4 | 8 | 16 |
---|---|---|---|---|---|
Runtime (s) | 4.03E + 00 | 1.81E + 00 | 9.50E − 01 | 5.29E − 01 | 2.77E − 01 |
Speedup | 1.00 | 2.22 | 4.24 | 7.62 | 14.55 |
5 Conclusions and future work
In this article, we propose a parallel algorithm for time fractional reaction-diffusion equation using the implicit finite-difference method. The algorithm includes a parallel solver for linear tridiagonal equations and parallel vector arithmetic operations. The solver is based on the divide-and-conquer principle and introduces a new tridiagonal reduced system with an elimination method. The experimental results shows the proposed parallel algorithm is valid and runs much more rapidly than the serial solution. The results also demonstrate the algorithm exhibits good scalability in performance. In addition, the introduced tridiagonal reduced system can be regarded as a general method for tridiagonal systems and applied on more applications. In the future, we would like to accelerate the solution of time fractional reaction-diffusion equation on heterogeneous architectures [35].
Declarations
Acknowledgements
This research work is supported in part by the National Natural Science Foundation of China under Grant Nos. 11175253, 61303265, 61402039, 91430218, 91530324, 61303265, 61170083 and 61373032, in part by Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20114307110001, in part by China Postdoctoral Science Foundation under Grant Nos. 2014M562570 and 2015T81127, and in part by 973 Program of China under Grant Nos. 61312701001 and 2014CB430205.
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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