Figure 2. Schematic diagram of the ASE-I scheme.

${G}_{{l}^{*}}^{\left(i\right)}={\left[\begin{array}{ccccc}1& -1& & & \\ -1& 2& -1& & \\ & \ddots & \ddots & \ddots & \\ & & -1& 2& -1\\ & & & -1& 1\end{array}\right]}_{{l}^{*}\times {l}^{*}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\left({l}^{*}=l\text{\hspace{0.05em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}l+2,\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}i=1,2,\cdots ,\frac{B-1}{2}\right),$

${Q}_{{l}^{\u2033}}$ is a zero matrix with ${l}^{\u2033}\times {l}^{\u2033}$ order ( ${l}^{\u2033}=l$ or $l-2$ ).

Using the properties of the function $g\left(x\right)={x}^{1-\alpha}\left(x\ge 1\right)$ , a set of conclusions can be obtained:

$\{\begin{array}{l}1={b}_{0}>{b}_{1}>\cdots \to 0,\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\displaystyle \underset{j=1}{\overset{k}{\sum}}{c}_{j}=1-{b}_{k}}\\ {\displaystyle \underset{j=1}{\overset{\infty}{\sum}}{c}_{j}=1,\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}1>2-{2}^{1-\alpha}={c}_{1}>{c}_{2}>\cdots \to 0}\end{array}$ (16)

2.3. Existence and Uniqueness of ASE-I Scheme Solution

Lemma 1. [17] Set $\rho >0$ , if matrix A is a non-negative real matrix, the matrix

${\left(I+\rho A\right)}^{-1}$ exists, and ${\Vert {\left(I+\rho A\right)}^{-1}\Vert}_{2}\le 1$ .

Lemma 2. The matrix ${G}_{1}$ and ${G}_{2}$ of ASE-I scheme are non-negative real matrix.

Proof. We only need to prove ${G}_{1}+{G}_{1}^{\text{T}}$ and ${G}_{2}+{G}_{2}^{\text{T}}$ are non-negative matrices. Obviously,

${G}_{{l}^{*}}^{\left(i\right)}+{\left({G}_{{l}^{*}}^{\left(i\right)}\right)}^{\text{T}}={\left[\begin{array}{ccccc}2& -2& & & \\ -2& 4& -2& & \\ & \ddots & \ddots & \ddots & \\ & & -2& 4& -2\\ & & & -2& 2\end{array}\right]}_{{l}^{*}\times {l}^{*}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{l}^{*}=l\text{\hspace{0.17em}}\text{or}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\left(l+2\right)\text{\hspace{0.05em}}$

are diagonally dominant matrices and their diagonal elements are non-negative

real numbers, therefore ${G}_{{l}^{*}}^{\left(i\right)}+{\left({G}_{{l}^{*}}^{\left(i\right)}\right)}^{\text{T}}$ are non-negative matrices. Similarly we can get ${\stackrel{\u02dc}{G}}_{l+1}^{\left(1\right)}+{\left({\stackrel{\u02dc}{G}}_{l+1}^{\left(1\right)}\right)}^{\text{T}}$ and ${\stackrel{\u02dc}{G}}_{l+1}^{\left(\frac{B+1}{2}\right)}+{\left({\stackrel{\u02dc}{G}}_{l+1}^{\left(\frac{B+1}{2}\right)}\right)}^{\text{T}}$ are also non-negative matrices.

According to Equation (15), ${\left(I+r{G}_{1}\right)}^{-1}$ and exists combing with Lemma 1 and Lemma 2. Then we can get

Theorem 1. The solution of the ASE-I scheme for solving time fractional sub-diffusion equation is uniquely solvable.

2.4. Stability of ASE-I Scheme

Lemma 3. Set, if A matrix is non-negative real matrix, we can get.

Proof. Because of

thus.

The growth matrix of ASE-I scheme for time fractional sub-diffusion equation is. Let

we can easily obtain by Lemma 3.

Suppose that is the solution of ASE-I scheme， is the approximate solution of the scheme，the error satisfies

(17)

in which. Let are the eigenvalues of respectively and the two matrices have the same eigenvalues. Hence

, , we can get

For,.

For, ,

,

Suppose that when, then we also have

Summing up we have. Hence, the following theorem is obtained.

Theorem 2. The ASE-I scheme for time fractional sub-diffusion equation is unconditionally stable.

2.5. Convergence of ASE-I Scheme

Because of

where,

hence we can get

At the same time the truncation error of improved Saul’yev asymmetric schemes used in spatial discretization is

, respectively.

When the two schemes are constructed alternatively on different time layers,

the truncation error is. In the same way, the truncation error of the explicit-implicit scheme (5-6) is

as well. Due to the first term of and can be cut off by the error term of temporal discretization, therefore the truncation error of ASE-I scheme is.

Define in which is exact solution of the Equation (1) and. Using to substitute into the Equation (15), we can get

in which

, C is a constant.

Lemma 4., , is a positive constant.

Proof . Lemma 4 can be proved using mathematical induction.

When,.

When,

Suppose that, then we also have

In summary, we have.

Due to, thus

where and T is a positive constant.

Theorem 3. The ASE-I scheme for time fractional sub-diffusion equation is unconditionally convergent and there is a positive A satisfies .

3. 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

(17)

in which the definition of and 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.

Theorem 4. The ASI-E scheme for time fractional sub-diffusion equation is unconditionally stable and convergent, meanwhile there is a positive A satisfies.

4. Numerical Examples

In this section, we present numerical examples to demonstrate that the ASE-I scheme is a computational effective numerical method for time fractional sub-diffusion equation compared with the implicit scheme as well as give the convergence rate of the ASE-I scheme. Numerical experiments will be done in MatlabR2015b, based on the Intel Core i5-2400 CPU@2.20GHz.

Considering the following time fractional sub-diffusion equation

The initial condition is

At, 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, when calculating numerical solutions, the computed results are listed in Table 1.

As these can be seen from Table 1, the numerical solutions of ASE-I and ASI-E scheme are better close to the exact solution compared with the implicit numerical solution, and the result of ASE-I scheme is obviously better, thus we mainly focus on ASE-I scheme. The surface of ASE-I numerical solution shown in Figure 3 describes the complete diffusion process which instructs the variations in concentration at different times and spaces. In particular, we consider the decay process curve of fractional diffusion model at the space point under the case of α taking different values. From Figure 4, it can be seen that the speed of diffusion is getting faster as α approaches to the number “1”, and the

Figure 3. Numerical solution surface of ASE-I scheme.

Table 1. Comparison of exact solution and numerical solutions.

Figure 4. Decay process curve of diffusion model.

diffusion velocity of solute becomes more and more slow with the reduction of the diffusion concentrations, this is consistent with measurements of some actual diffusion processes.

Next, to better validate the stability and compare the accuracy of the ASE-I, we will analyze the change cure of the sum of relative error with time steps (SRET) and the distribution of the difference total energy (DTE) at space grid points. Taking the exact solution as the control solution, we let the numerical solution of the scheme as the perturbation solutions. The definition of SRET and DTE are as follows:

From Figure 5, the SRET of ASE-I scheme is less than 5. The relative error is a little big in the first few steps, and decreases rapidly with the time step, thus we can know that the ASE-I scheme of the time fractional sub-diffusion equation is stable.

The values of ASE-I scheme’s DTE are between 0 and 0.0075 from Figure 6, and the calculation error of the diffusion concentration is getting reductive with the passage of space, this also can demonstrate that the ASE-I scheme of time fractional sub-diffusion equation is very close to the exact solution. The values of DTE appear to fluctuate near the grids 16, 32, 48 64, and its maximum values appear near the grids 16 and 32. Fortunately these grids are the “inter boundary point” of the ASE-I scheme, i.e. a couple of Saul’yev scheme alternatively applied in different temporal level. At the same time, the explicit-implicit scheme is applied in the “inter point” for the ASE-I scheme. So it is normal to conclusion that the values of DTE in “inter boundary point” are little bigger than that in “inter point”. Moreover we compare the DTE of the implicit scheme and ASE-I

Figure 5. Change curve of SRET.

Figure 6. Distribution of DTE at space points.

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 2 gives the computational errors with different temporal step sizes using the fractional order. We can see that the numerical accuracy in temporal direction is approximately order and compared with the implicit scheme, it has the higher accuracy. At the same time, we compute the numerical accuracy in spatial direction. Taking for Implicit and ASE-I scheme and let. From 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 and as the spatial grid number and the temporal grid number. In terms of computation time in Table 4, the computational efficiency (CPU time) of the ASE-I (ASI-E) scheme has big advantage compared with implicit scheme. With the increase of the grid number, the computation times of the implicit scheme rapidly grow up, and which the ASE-I (ASI-E) scheme’s has a lower growth rate by comparison. The computation time of the ASE-I (ASI-E) scheme can save nearly 75% compared with the implicit scheme and the Sp (Speedup) is approximately 4.38 and 4.16 respectively. Comprehensively considering the computing efficiency and the computing accuracy, the ASE-I scheme can be more effective to solve the time fractional sub-diffusion equation. When the long time course is calculated, the parallel computing advantages of ASE-I scheme will be more obvious.

Table 2. Numerical errors and convergence of ASE-I scheme in temporal direction (h = 0.02).

Table 3. Numerical errors and convergence of ASE-I scheme in spatial direction ().

Table 4. Comparison of the three difference schemes’ CPU time.

5. Conclusion

For the time fractional sub-diffusion equation, this paper constructs the ASE-I and ASI-E difference schemes with unconditional stability and convergence. Numerical experiments verify the theoretical analyses and show that the proposed scheme is of excellent computational accuracy and obvious parallel properties. The ASE-I (ASI-E) scheme given by this paper can be extended to solve other fractional diffusion models and the parallel computing advantages of the ASE-I (ASI-E) scheme will be more obvious for the long time course and the high dimensional fractional diffusion equation. But the application of the ASE-I (ASI-E) scheme in multidimensional fractional differential equations remains to be further studied.

Acknowledgements

The research is supported by the National Natural Science Foundation of China (Grant No. 11371135) and the Fundamental Research Funds of the Central Universities (Grant No. 2018MS168).

Conflicts of Interest

The authors declare no conflicts of interest.

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