A Class of Semi-Implicit Parallel Difference Method for Time Fractional Diffusion Equations

In this paper, we construct a class of semi-implicit difference method for time fractional diffusion equations—the group explicit (GE) difference scheme, which is a difference scheme with good parallelism constructed using Saul’yev asymmetric scheme. The stability and convergence of the GE scheme of time fractional diffusion equation are analyzed by mathematical induction. Then, the theoretical analysis is verified by numerical experiments, which shows that the GE scheme is effective for solving the time fractional diffusion equation.


Introduction
The fractional anomalous diffusion model has a profound physical background and rich theoretical connotation. It is widely used in the fields of fluid mechanics, signal processing and information recognition, fractal theory, etc. It has become an important tool for describing various complex mechanical behaviors [1] [2] [3]. However, the analytical solutions of fractional differential equations are mostly difficult to give explicitly. It is necessary and important to study the numerical solution of fractional differential and integral equations [4] [5] [6].
Zhuang and Liu [13] constructed a class of implicit difference schemes with unconditional stability and convergence for time fractional diffusion equations. with the extrapolation technique. Gao and Sun [16] gave a compact difference scheme for the slow diffusion equation, which has fourth-order accuracy in space.
However, due to the memory and non-locality of fractional derivatives, the computational and storage quantities of numerical calculations of fractional differential equations are very large. When we simulate practical problems, the requirements of computational resources will be very high. However, the existing serial algorithm has a large amount of calculation and relatively low computational efficiency. We mainly study the fast numerical algorithm of fractional differential equation to improve the numerical simulation efficiency of fractional order modeling this paper.
With the rapid development of multi-core and cluster technology, parallel algorithms have become one of the mainstream technologies to improve the efficiency of numerical calculations [17] [18]. Zhang and Gu [19] proposed a piecewise implicit scheme for the integer order diffusion equations in an asymmetrical scheme, and used alternating techniques to construct multiple explicitimplicit and implicit alternating parallel methods. This kind of parallel method has been widely used in integer order evolution equations. For fractional differential equations, Gong and Bao [20] [21] performed parallel computation on the explicit difference schemes of fractional reaction-diffusion equations. The core content of their parallelization is parallel calculation of matrix and vector product, vector and vector addition. Sweilam and Moharram [22] constructed a parallel C-N scheme for time fractional parabolic equations. The core of the method is to solve the equation Ax b = using the preconditioned conjugate gradient method. We do not study the parallel algorithm of equations from the perspective of numerical algebra, but based on the parallelization of traditional differential schemes for solving fractional diffusion equations numerically [23].
For the time fractional diffusion equation, the Saul'yev asymmetric scheme is given, and then the Saul'yev asymmetric scheme is used to construct the group explicit (GE) scheme of the time fractional diffusion equation with parallel nature. The mathematical induction method is used to prove that the GE of the time fractional diffusion equation is unconditionally stable and convergent. Finally, the theoretical analysis is verified by numerical experiments, which shows that the GE scheme is very effective for solving time fractional diffusion equations.

Fractional Diffusion Equation
We consider the initial-boundary value problem of the fractional diffusion equation Journal of Applied Mathematics and Physics with the initial conditions (2) and the boundary conditions

Construction of GE Scheme for Fractional Diffusion Equation
Construct the Saul'yev asymmetric scheme of Equation (1) above two types of Saul'yev asymmetric schemes can be rewritten as follows: The GE scheme of the time fractional diffusion equation is designed as follows: the four "○" in Figure 1 indicate the establishment of the Saul'yev asymmetric format (10) at point ( ) , 1 i k + , four "□" means that the Saul'yev asymmetric format (9) is established at point ( ) scheme is a combination of these two asymmetric schemes, and the difference equation is the following 2 2 × equations.
where ( ) From Equations (13) and (14), we can know that the value of two points  (9) and (10) is called a group explicit method, or GE method, and this method is easy to calculate in parallel.
According to the parity of the number of points m, the GE method has different forms.

1) GER method
It has a right single point and ( ) The GER method can be expressed as the following matrix form:

2) GEL method
It has a left single point, except that the value of the point ( ) When the number of points m is odd, and 1 m − is even, there are ( )

4) GEC method
It includes ( ) Using the properties of function ( ) ( ) 1 1 g x x x α − = ≥ , we can draw the following conclusions:

Stability Analysis of GE Scheme
In this section, we analyze the stability of the GE scheme. Taking GER scheme as an example, the stability of GER scheme is analyzed. Assume In summary, we have

Numerical Experiment
In this section, we will compare and analyze the analytic solutions of GE scheme and the classical implicit scheme by numerical examples. It shows that GE scheme in this paper is effective for solving time fractional diffusion equations.
GE scheme can also be applied to other types of time fractional equations.
Consider the following time fractional diffusion equation [13]: The boundary conditions are: ( ) ( )  Table 1, whose curves are show in Figure 2.
It can be seen from Figure 2 that the surface of GE scheme solution is smooth, which is same as the surfaces of analytical solution and the implicit scheme solution, which shows that GE scheme is feasible to solve the fractional diffusion equation. In terms of accuracy, it can be seen from Table 1 that the difference scheme solution is very close to the analytical solution (in the case of the first 20 terms). It can be seen that difference scheme is effective for solving time fractional diffusion equation. In terms of calculation time (CPU time), it can be seen that when the first 20 terms of the analytical solution formula are taken, the calculation time is 21.5483 s, which shows that the calculation amount is very large.
The computation time of the two difference schemes is less than 1 s. It can be seen that the difference method is effective numerical methods for solving the time fractional diffusion equation.
By comparing GE scheme of this paper with the analytical solution and the implicit difference scheme, it can be seen from Table 1 that the absolute error of the numerical solution of two schemes is between 10 −3 -10 −4 , but the calculation amount (CPU time) of GE scheme given in this paper is only 57.1% of implicit difference scheme. Because GE schemein this paper has the property of parallel computing, compared with the implicit difference scheme, GE scheme of the fractional diffusion equation improves the computing efficiency by about 43% when the calculation accuracy is equivalent. When performing long term calculations, the advantages of parallel computing in GE scheme will be more obvious. Table 1. Compare and analysis of two difference scheme solutions and analytical solutions.   Table 2 and Table 3. From Table 2, we can see that GE scheme converges linearly in the time direction. From Table 3, we can see that GE scheme converges squarely in the spatial direction. The numerical results are consistent with the theoretical analysis.

Conclusion
In this paper, the group explicit (GE) scheme of the time fractional diffusion equation is constructed by applying the Saul'yev asymmetric scheme. We analyzed the stability and convergence of GE scheme. GE scheme has the property of parallel computing, and its computation efficiency is nearly 60% less than that of the classic implicit scheme. The numerical experimental results are consistent with the theoretical analysis. GE scheme has 2 α − order of convergence in time and second order of convergence in space. Theoretical analysis and numerical experiments show that GE scheme is effective for solving time fractional diffusion equations. Especially for long term history and large computational domain problems, the advantages of GE scheme for parallel computing will be more obvious.