The Splitting Crank–Nicolson Scheme with Intrinsic Parallelism for Solving Parabolic Equations

In this paper, a splitting Crank–Nicolson (SC-N) 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 n -th 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 two-dimensional parabolic equations by alternating direction implicit (ADI) technique. Numerical experiments illustrate the accuracy and ef-ﬁciency of the new algorithm.

On this basis, a new parallel algorithm called splitting Crank-Nicolson scheme for parabolic equations will be presented in this paper. e 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. en, 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 n-th 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 n-th time level. (3) e value U n+1 � (V n+1 + W n+1 )/2 are set as numerical solutions at (n + 1)th time level.
e advantage of the SC-N scheme is spitting the C-N scheme into two parallel algorithms, which can be computed by parallel computers. en, the average value obtained is restored to C-N scheme approximately. is paper is organized as follows: in Section 2, we introduce a SC-N scheme for parabolic equations. For simplicity of presentation, we focus on a model problem, namely, one-dimensional parabolic equations. e new algorithm and detailed presentations are given. en, we extend the new algorithm to solve two-dimensional parabolic equations by ADI technique. Finally, numerical experiments illustrated the accuracy of SC-N scheme is approximate to the C-N scheme and the new algorithm is efficient.

Algorithm Presentation
Considering the model problem of one-dimensional parabolic equations, with the initial and boundary conditions where 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 u n j be the approximate solution at (x j , t n ). u (x, t) represents the exact solution of (1). e well-known Crank-Nicolson scheme can be written as where r � τ/h 2 . Let μ � ar, and (3) can be written as the matrix form where U n � (u n 1 , u n 2 , . . . , u n m−1 ) T and F n � ((μ/2)(u n 0 + u n+1 0 ) + τf n 1 , τf n 2 , . . . , τf n m−2 , (μ/2)(u n m + u n+1 m ) + τf n m−1 ) T . e matrices A and B are as follows: Splitting Crank-Nicolson scheme (4), we obtain where 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. en, Scheme (8) is DDM I, and scheme (9) is called DDM II. e 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: −aru n+1 e flow chart of schemes (11)- (14) are displayed in Figure 1.
Assume m − 1 � 6K, where K is a positive integer.
For the values u n+1 m−3 , u n+1 m−2 , u n+1 m−1 by using the formulas as follows: Journal of Function Spaces 3 DDM I can be written as the matrix form where U n � (u n 1 , u n 2 , · · · , u n m−1 ) T , A 1 � (I + μG 1 ), and e 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.
Find the values u n+1 m−6 , u n+1 m−5 , . . . , u n+1 m−1 by using the formulas as follows: DDM II can be written as the matrix form where A 2 � (I + μG 2 ) and B 2 � (I − μG 1 ). e 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.

Remark 1. Obviously, C-N scheme (4) is equal to
Remark 2. Because the SC-N scheme is derived from the C-N scheme, it also has the properties of the C-N scheme, i.e., the SC-N scheme is unconditionally stable and maintains the second-order numerical accuracy O (τ 2 + h 2 ).

Extension to Two-Dimensional 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 u n i,j 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).

x-Direction.
Let r 1 � aτ/(2h 2 ) and r 2 � bτ/(2h 2 ), and the matrix form of the SC-N scheme in x-direction can be written as follows:

y-Direction.
e matrix form of the SC-N scheme in ydirection can be written as follows: Require: Initialization U 0 (x j ) ⟵ u 0 (x j ). for n � 0, 1, · · ·, N do for j � 0, 1, · · ·, m do Solve the values V n+1 j by using DDM I (18). Solve the values W n+1 j by using DDM II (24). e average of two values will be calculated, i.e. U n+1 . end for end for Ensure: Output U N (x j ).
e corresponding algorithm can be described in Algorithm 2.
Similar to Algorithm 1, it is obvious that Algorithm 2 has unconditional stability and parallelism.
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 x-direction by (30) at the halftime step and then solve the values along y-direction by (32) at the next half-time 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.
ough it is developed for two-dimensional problems, Algorithm 2 can be easily extended to solve high-dimensional parabolic equations.

Numerical Experiments
To illustrate the efficiency of the SC-N scheme for parabolic equations, we will compare the accuracy of the new algorithm with the existing method.
e exact solution of Example 1 is Firstly, 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 L ∞ � ‖u(x j , t n ) − u n j ‖ ∞ taking τ � 0.001, and the following rate of convergence in the space is Clearly, the errors appear to be of order O (h 2 ) in Table 1. Next, we present the error results of the SC-N scheme in terms of the absolute errors and the relative errors, where the absolute error (A. E.) is defined by e n j � u n j − u x j , t n , (37) and the relative error (R. E.) is defined by Tables 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 ASC-N scheme in [3] and C-N 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 ASC-N 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 C-N 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.

Conclusion
We have proposed and analyzed the SC-N scheme for parabolic equations. is algorithm consists of two DDMs, which split the Crank-Nicolson scheme; each one is used to solve the values at the same time level. en, the average of two values is calculated. e SC-N scheme maintains the properties of the C-N scheme, i.e., unconditional stability and second-order numerical accuracy. en, we extend the new algorithm to two-dimensional parabolic equations by ADI technique, which means that high-dimensional 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
e authors declare that they have no conflicts of interest.