Alternating Segment Explicit-Implicit and Implicit-Explicit Parallel Difference Method for Time Fractional Sub-Diffusion Equation

The fractional diffusion equations can accurately describe the migration process of anomalous diffusion, which are widely applied in the field of natural science and engineering calculations. This paper proposed a kind of numerical methods with parallel nature which were the alternating segment explicit-implicit (ASE-I) and implicit-explicit (ASI-E) difference method for the time fractional sub-diffusion equation. It is based on the combination of the explicit scheme, implicit scheme, improved Saul’yev asymmetric scheme and the alternating segment technique. Theoretical analyses have shown that the solution of ASE-I (ASI-E) scheme is uniquely solvable. At the same time the stability and convergence of the two schemes were proved by the mathematical induction. The theoretical analyses are verified by numerical experiments. Meanwhile the ASE-I (ASI-E) scheme has the higher computational efficiency compared with the implicit scheme. Therefore it is feasible to use the parallel difference schemes for solving the time fractional diffusion equation.


Introduction
Fractional differential equations arise from some anomalous diffusion models and can be very useful in describing the memory and heritability of various complex substances. Because of its deep physical background and rich theoretical significance, it has been widely used in various fields such as fluid mechanics, signal processing and information recognition [1] [2] [3]. Due to the success in the analysis of a discrete non-Markovian Random Walk Approximation for the Journal of Applied Mathematics and Physics unconditional stability and convergence are analyzed. In Section 3, we give the alternating segment implicit-explicit (ASI-E) parallel difference method. In Section 4, numerical experiments are presented to support our theoretical analysis and indicate that the ASE-I (ASI-E) scheme is effective for solving time fractional sub-diffusion equation.

Time Fractional Sub-Diffusion Equation
The fractional sub-diffusion equation is considered as follows x (1) Initial boundary conditions: , u x t indicates the diffusion concentration in point x at time t, fractional order derivative α in Equation (1) is Caputo fractional derivative defined by By taking the finite sine transform and Laplace transform, the exact solution for the Equation (1) with the boundary conditions as above is obtained as Equation (3)

Construction of ASE-I Scheme
Second, the classical explicit scheme is At last, we present the two improved Saul'yev asymmetric schemes ( ) ( ) ( ) ( ) we can respectively derive the four schemes of Equation (1): As the scheme we constructed above, the classical implicit scheme (5) is absolutely stable but it is inconvenience to efficiently obtain the results because of needing to solve three diagonal matrixes. The classical explicit scheme (6) has ideal parallelism but it is conditionally stable. The improved Saul'yev asymmetric schemes (7) and (8) are convenient to parallel computing, but are conditionally stable. So the ASE-I scheme which we constructed is combined with the advantages of the above schemes and the design is as follows Let 1 m Bl − = , here B is a positive odd number, w is a positive integer, We divide the points on each time level into B sections, order as 1 2 , , , B S S S  . And on the even level, we arrange the computation according to the rule of "the explicit segment-the implicit segment-the explicit segment". When it turns to the odd level, the rule changes into "the implicit segment-the explicit segment-the implicit segment" that makes the implicit segment and the Journal of Applied Mathematics and Physics In order to improve the calculation accuracy, the left and right boundary point of implicit segment will be replaced by the implicit scheme when 0 0 i =   We use ○ to denote the classical explicit scheme, • to denote the classical implicit scheme, the remainder are two improved asymmetric formats. Thus the ASE-I scheme can be written as follows 0, 2, Using the properties of the function ( ) ( ) , a set of conclusions can be obtained: Proof. We only need to prove

Existence and Uniqueness of ASE-I Scheme Solution
are also non-negative matrices.

Stability of ASE-I Scheme
Proof. Because of ( )( ) The growth matrix of ASE-I scheme for time fractional sub-diffusion equation , rG rG respectively and the two matrices have the same eigenvalues. Hence , we can get

Convergence of ASE-I Scheme
Because of ( ) (      ( ) Proof . Lemma 4 can be proved using mathematical induction.

ASI-E Parallel Difference Method
Imitating the method constructed ASE-I scheme, we give the ASI-E scheme for solving the time fractional sub-diffusion equation. The difference between the ASE-I and ASI-E scheme is that the use of implicit segment and explicit segment is different.
On the odd level, we arrange the computation according to the rule of "the implicit segment-the explicit segment-the implicit segment", when it turns to the even level, the rule changes into "the explicit segment-the implicit segment-the explicit segment". Thus we get the ASI-E difference scheme 0, 2, in which the definition of 1 2 , C C and k b are the same as above. Due to the implicit scheme on the first layer is unconditionally stable and convergent, we imitate the analytical and proved method of the ASE-I scheme (15) from the second time layer, and get the following theorem.

Numerical Examples
In The initial condition is ( ) we compare the solution of ASE-I scheme with the exact solution and the numerical solution using the implicit scheme. For the exact solution, the series in Equation (3) is truncated after 20 terms. We take 1000 n = , 80 m = when calculating numerical solutions, the computed results are listed in Table 1.
As these can be seen from Table 1 Figure 4, it can be seen that the speed of diffusion is getting faster as α approaches to the number "1", and the    scheme. As shown in Figure 6, there is not much difference between the two schemes and the error of the ASE-I scheme is slightly smaller than the other one.
Comprehensively considering the ASE-I scheme can be more effective to solve the time fractional sub-diffusion equation.
A test example will be performed to illustrate the convergence order of the ASE-I scheme. Denote [22] ( ) Thus the numerical results are presented as follows.  Table 3, we can see that the numerical accuracy in spatial direction is second-order for Implicit and ASE-I scheme, therefore the experimental results are basically consistent with the theoretical analysis.
At last we select 1000 m = and 200, 400, 600,800,1000,1200 n = as the spatial grid number and the temporal grid number. In terms of computation time in Table 4