Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020
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Guanyu Xue, Yunjie Gong, Hui Feng, "The Splitting Crank–Nicolson Scheme with Intrinsic Parallelism for Solving Parabolic Equations", Journal of Function Spaces, vol. 2020, Article ID 8571625, 12 pages, 2020. https://doi.org/10.1155/2020/8571625
The Splitting Crank–Nicolson Scheme with Intrinsic Parallelism for Solving Parabolic Equations
Abstract
In this paper, a splitting Crank–Nicolson (SCN) scheme with intrinsic parallelism is proposed for parabolic equations. The new algorithm splits the Crank–Nicolson scheme into two domain decomposition methods, each one is applied to compute the values at (n + 1)th time level by use of known numerical solutions at nth time level, respectively. Then, the average of the above two values is chosen to be the numerical solutions at (n + 1)th time level. The new algorithm obtains accuracy of the Crank–Nicolson scheme while maintaining parallelism and unconditional stability. This algorithm can be extended to solve twodimensional parabolic equations by alternating direction implicit (ADI) technique. Numerical experiments illustrate the accuracy and efficiency of the new algorithm.
1. Introduction
The parallel numerical schemes for parabolic equations have been studied rapidly. In 1985, Evans [1] proposed the Alternating Group Explicit (AGE) scheme for the diffusion equation by Saul’yev asymmetric schemes [2]. The Alternating Segment Crank–Nicolson (ASCN) scheme was designed in [3]. Afterwards, the alternating segment algorithms (AGE scheme and ASCN scheme) above became very effective methods for parabolic equations, such as convectiondiffusion equation [4], dispersive equation [5–8], and fourthorder parabolic equation [9, 10]. Meanwhile, domain decomposition methods (DDMs) for the partial differential equations have been developed. Since the advantages of the low computation and communication costs at each time step, the nonoverlapping DDMs have been studied extensively [11–20]. The concept called “intrinsic parallelism” was presented in [21–23]. In 1999, the alternating difference schemes were presented, and the unconditional stability analysis was given in [24]. The unconditionally stable domain decomposition method was obtained by the alternating technique in [25–30]. Recently, the numerical methods for parabolic equations have attracted great attention of scholars [31–42]. Meanwhile, the finite volume element methods for elliptic problems are studied by Bi et al. [43, 44], and the finite volume element method for secondorder hyperbolic equations is proposed by Chen et al. [45]. Obviously, the Crank–Nicolson scheme is one of the most classical methods for PDEs [46, 47]. But, there is litter work on parallel algorithms that satisfy the accuracy and stability of the Crank–Nicolson scheme.
On this basis, a new parallel algorithm called splitting Crank–Nicolson scheme for parabolic equations will be presented in this paper. The idea of the new algorithm is to divide the classical Crank–Nicolson scheme into two parts, which are DDMs, each of which is used in computations at (n + 1)th time level utilizing the numerical solution at time level n. Then, the average of the above two values is chosen to be the numerical solutions at (n + 1)th time level. It can be described as follows:(1)Split the Crank–Nicolson scheme into DDM I and DDM II by Saul’yev asymmetric schemes.(2)DDM I is applied to compute the values at (n + 1)th time level noted as V^{n+1} by use of known value U^{n} at nth time level. DDM II is also used to compute the values at (n + 1)th time level noted as W^{n+1} by use of U^{n} at nth time level.(3)The value U^{n+1} = (V^{n+1} + W^{n+1})/2 are set as numerical solutions at (n + 1)th time level.
The advantage of the SCN scheme is spitting the CN scheme into two parallel algorithms, which can be computed by parallel computers. Then, the average value obtained is restored to CN scheme approximately. This paper is organized as follows: in Section 2, we introduce a SCN scheme for parabolic equations. For simplicity of presentation, we focus on a model problem, namely, onedimensional parabolic equations. The new algorithm and detailed presentations are given. Then, we extend the new algorithm to solve twodimensional parabolic equations by ADI technique. Finally, numerical experiments illustrated the accuracy of SCN scheme is approximate to the CN scheme and the new algorithm is efficient.
2. Algorithm Presentation
Considering the model problem of onedimensional parabolic equations,with the initial and boundary conditionswhere a > 0 is a constant.
Let h and τ be the spatial and temporal step sizes, respectively. Denote x_{j} = jh, j = 0, 1, …, m and t_{n} = nτ, n = 0, 1, …, N. Let be the approximate solution at (x_{j}, t_{n}). u (x, t) represents the exact solution of (1).
The wellknown Crank–Nicolson scheme can be written aswhere r = τ/h^{2}.
Let μ = ar, and (3) can be written as the matrix formwhere and .
The matrices A and B are as follows:
Splitting Crank–Nicolson scheme (4), we obtainwhere A_{1} and A_{2} are block diagonal matrices, respectively.
V ^{n+1} and W^{n+1} can be defined as follows:where V^{n+1} and W^{n+1} approximate to U^{n+1}, respectively. Then,
Scheme (8) is DDM I, and scheme (9) is called DDM II. The two DDMs aforementioned are suitable for parallel computing.
In order to construct two domain decomposition methods, DDM I and DDM II, at (n + 1)th time level, we need to consider four forms of Saul’yev asymmetric difference schemes corresponding to (3) as follows:
The flow chart of schemes (11)–(14) are displayed in Figure 1.
(a)
(b)
(c)
(d)
Assume m − 1 = 6K, where K is a positive integer.
2.1. DDM I
For the values , we use the formulas as follows:
Find the values by using the following formulas (k = 1, 2, …, K − 1):
Obviously, each subdomain contains 6 nodes which can be computed by (16) independently.
For the values by using the formulas as follows:
DDM I can be written as the matrix formwhere , A_{1} = (I + μG_{1}), and B_{1} = (I − μG_{2}).
The matrices G_{1} and G_{2} are block diagonal matrices which are as follows:where
Each block matrix system (i.e., each subdomain) can be solved independently. It is evident that DDM I (18) has intrinsic parallelism.
2.2. DDM II
For the values by using the following formulas:
Find the values by using the formulas as follows (k = 1, 2, …, K − 2):
Obviously, each subdomain contains 6 nodes which can be computed by (22) independently.
Find the values by using the formulas as follows:
DDM II can be written as the matrix formwhere A_{2} = (I + μG_{2}) and B_{2} = (I − μG_{1}).
The matrices G_{1} and G_{2} are block diagonal matrices which are as follows:
Each block matrix system (i.e., each subdomain) can be solved independently. It is evident that DDM II (24) has intrinsic parallelism.
Schemes (15)–(17) and schemes (21)–(23) construct two domain decomposition methods (18) and (24), respectively. The corresponding algorithm can be described in Algorithm 1. The matrix form of Algorithm 1 can be written as follows:where , , and .
Remark 1. Obviously, CN scheme (4) is equal to .
Remark 2. Because the SCN scheme is derived from the CN scheme, it also has the properties of the CN scheme, i.e., the SCN scheme is unconditionally stable and maintains the secondorder numerical accuracy O (τ^{2} + h^{2}).
3. Extension to TwoDimensional Parabolic Equations
In this section, we will extend Algorithm 1 to solve twodimensional parabolic equations:where domain Ω ∈ (0, L_{x}) × (0, L_{y}) and a > 0 and b > 0 are diffusion coefficients.
Let be the approximate solution at (x_{i}, y_{j}, t_{n}), and u (x, y, t) represents the exact solution of (27). With the same time and space discretization of Algorithm 1, we obtain its extended algorithm by alternating direction implicit (ADI) technique [48] for equations (27)–(29).
3.1. xDirection
Let r_{1} = aτ/(2h^{2}) and r_{2} = bτ/(2h^{2}), and the matrix form of the SCN scheme in xdirection can be written as follows:where
3.2. yDirection
The matrix form of the SCN scheme in ydirection can be written as follows:where
The corresponding algorithm can be described in Algorithm 2.
Similar to Algorithm 1, it is obvious that Algorithm 2 has unconditional stability and parallelism.
Remark 3. In fact, is the CN scheme in xdirection, and is the CN scheme in ydirection.
Remark 4. In Algorithm 2, the domain is divided into many subdomains by using two DDMs. In each time interval, we first solve the values along xdirection by (30) at the halftime step and then solve the values along ydirection by (32) at the next halftime step. Schemes (30) and (32) lead to block diagonal algebraic systems that can be solved independently. So, Algorithm 2 not only is suitable for parallel computation but also maintains the accuracy. Based on the advantage of ADI technique, Algorithm 2 reduces computational complexities. Though it is developed for twodimensional problems, Algorithm 2 can be easily extended to solve highdimensional parabolic equations.
4. Numerical Experiments
To illustrate the efficiency of the SCN scheme for parabolic equations, we will compare the accuracy of the new algorithm with the existing method.
Example 1. The exact solution of Example 1 isFirstly, we examine the convergence rate of Algorithm 1. We divide the mesh point into many segments, such as K = 3, K = 4, K = 5, and K = 6. We calculate errors taking τ = 0.001, and the following rate of convergence in the space isClearly, the errors appear to be of order O (h^{2}) in Table 1.
Next, we present the error results of the SCN scheme in terms of the absolute errors and the relative errors, where the absolute error (A. E.) is defined byand the relative error (R. E.) is defined byTables 2 and 3 display the absolute errors and the relative errors obtained by presented Algorithm 1 for h = 1/19 (i.e., K = 3) and h = 1/25 (i.e., K = 4) at t = 0.2, t = 0.4, and t = 0.8 when taking r = 1.5 (r = τ/h^{2}). From Tables 2 and 3, it is obvious that our algorithm has high accuracy.
We compare Algorithm 1 with the ASCN scheme in [3] and CN scheme (3) by the maximum errors for h = 1/19 (i.e., K = 3) and h = 1/25 (i.e., K = 4) at different times t = 0.2, t = 0.4, t = 0.6, and t = 0.8. With the increasing of computation time, the errors of the ASCN scheme in [3] increase more than those of Algorithm 1 for different r (r = τ/h^{2}) in Tables 4 and 5. Moreover, the results show that Algorithm 1 can achieve the same accuracy as the classic CN scheme while maintaining parallelism. We consider an example for h = 1/121 (i.e., K = 20) with large grid ratio r = 15 (r = τ/h^{2}). Table 6 shows that Algorithm 1 has a better accuracy than two others, and it is indicated that Algorithm 1 is stable.





Example 2. where the domain is Ω = (0, 1) × (0, 1), and the right side function isThe corresponding exact solution isLet h = 1/19 (i.e., K = 3), h = 1/25 (i.e., K = 4), h = 1/31 (i.e., K = 5), and h = 1/37 (i.e., K = 6), and the maximum errors of Algorithm 2 for τ = 0.0004 at different times are displayed in Table 7. It is obvious that the accuracy of Algorithm 2 is satisfied. Table 8shows the comparison of the CPU calculation time among Algorithm 2, the classical CN scheme, and the ASCN scheme. Algorithm 2 not only has high accuracy but also has good parallel efficiency.


5. Conclusion
We have proposed and analyzed the SCN scheme for parabolic equations. This algorithm consists of two DDMs, which split the Crank–Nicolson scheme; each one is used to solve the values at the same time level. Then, the average of two values is calculated. The SCN scheme maintains the properties of the CN scheme, i.e., unconditional stability and secondorder numerical accuracy. Then, we extend the new algorithm to twodimensional parabolic equations by ADI technique, which means that highdimensional parabolic equations can be solved by the proposed algorithm in this paper. Numerical experiments illustrate the good performance of the new algorithm.
Data Availability
All data generated or analyzed during this study are included in this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the anonymous referees for helpful comments and suggestions. This research was supported by the PhD research startup foundation of Yantai University (no. 2219002).
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