A Novel High-Order Finite-Difference Method for the Time-Fractional Diffusion Equation with Smooth/Nonsmooth Solutions

Abstract

A time-fractional diffusion equation with variable coefficients and Caputo derivative of order $\alpha \in (0,1)$ is considered. Recently, some S-type, i.e., S1, S2, and S3, formulae have been established with high-order accuracy for approximating the Caputo derivative. These formulas are based on B-splines of degree l with the global $(l+1-\alpha )$-order accuracy. On the other hand, the typical solution of such diffusion equations has weak regularity near the initial time. In this paper, we aim to establish a new finite-difference method based on the transformed S2 discretization, called the S2-FD method, dealing with this singularity of the solution. We analyze the stability and convergence of the proposed S2-FD scheme for some problems with smooth/nonsmooth solutions. We also indicate the stability of the classic S2-FD method for smooth solutions. It is also proved that both have the global convergence of order $3-\alpha$. The obtained results are confirmed by some numerical examples.

The Classic S2-FD Method

Let $T=1$, n be a positive integer, $\Delta t=1/n$, and $t_k = k\, \Delta t$ for $k=0,1,\ldots ,n$ denotes a uniform mesh on [0, 1] . We approximate ${\mathcal {D}}^{\alpha }_t u(t)$ using the classic S2 formula [28]

$$\begin{aligned} {\mathcal {D}}^{\alpha }_t u(t)\big \vert _{t = t_k}&=\dfrac{1}{\Gamma (1-\alpha )} \sum _{j=1}^{k} \int _{t_{j-1}}^{t_j} \dfrac{u'(s)}{(t_k - s)^\alpha } \mathrm{d}s \nonumber \\&\simeq \dfrac{1}{\mu } \Big ( \beta ^{k+1} - \sum _{j=1}^{k} (c^{ (\alpha )}_{k-j} - c^{ (\alpha )}_{k-j+1}) \, \beta ^{j} - c_{k}^{(\alpha )} \, \beta ^{0} \Big ) := D ^{\alpha , 2}_t u(t)\big \vert _{t = t_k}, \end{aligned}$$

(A.1)

where $\mu = \Delta t^{\alpha } \Gamma (3-\alpha )$ and

$$\begin{aligned} c^{(\alpha )}_j := \left\{ \begin{array}{ll} a^{(\alpha )}_{0} - b^{(\alpha )}_{0}, &{} j=0, \\ a^{(\alpha )}_{j} - a^{(\alpha )}_{j-1}, &{} 1\le j \le k-1, \\ b^{(\alpha )}_{k} - a^{(\alpha )}_{k-1}, &{} j=k, \end{array} \right. \end{aligned}$$

(A.2)

with $a^{(\alpha )}_{j} := (j+1)^{2-\alpha }-j^{2-\alpha }$ and $b^{(\alpha )}_{j} := (2-\alpha ) \, j^{1-\alpha }$. Some properties of $c_j^{(\alpha )}$’s defined in (A.2) are mentioned in the next Lemma.

Lemma A.1

For $k \ge 1$, we have

• 1. $c_j^{(\alpha )} > 0$ for $j=0,1,\ldots ,k$,

• 2. $c_0^{(\alpha )} \ge c_{k}^{(\alpha )}$,

• 3. $c_1^{(\alpha )} \ge c_2^{(\alpha )} \ge c_3^{(\alpha )} \ge \cdots \ge c_{k}^{(\alpha )}$,

• 4. $c_j^{(\alpha )} \leqslant 2$ for $j=1,\ldots ,k$.

Proof

Noticing $a_{j}^{(\alpha )} = (2-\alpha ) \int _{j}^{j+1} x^{1-\alpha } dx$ and $x^{1-\alpha }$ being strictly increasing with respect to x, it is easy to verify that

$$\begin{aligned} 1= a_{0}^{(\alpha )}< a_{1}^{(\alpha )} <\cdots . \end{aligned}$$

Hence, $c_j^{(\alpha )} > 0$ for $j=0,\ldots , k-1$. On the other hand,

$$\begin{aligned} c_{k}^{(\alpha )} = b_{k}^{(\alpha )} - a_{k-1}^{(\alpha )} = (2-\alpha ) k ^{1-\alpha }- k ^{2-\alpha } + (k-1) ^{2-\alpha }. \end{aligned}$$

Let $g(x) = (2-x) k ^{1-x}- k ^{2-x} + (k-1) ^{2-x}$, then $g'(x) <0$ for $x\in (0,1)$. Therefore, g is monotone decreasing on (0, 1) and $0 = c_{k}^{(1)}< c_{k}^{(\alpha )} < c_{k}^{(0)} = 1 = c_{0}^{(\alpha )}$. For the third part, letting $h(x) = x^{2-\alpha }$ follows that

$$\begin{aligned} h'''(x) = (2 - \alpha ) (1-\alpha ) (-\alpha ) x^{-1-\alpha } < 0. \end{aligned}$$

Eventually, $c_j^{(\alpha )} - c_{j+1}^{(\alpha )} = -h(j+2) + 3 h(j+1) - 3 h(j) + h(j-1) = -h'''(\xi _j) > 0$ which leads to $c_1^{(\alpha )} \ge c_2^{(\alpha )} \ge c_3^{(\alpha )} \ge \cdots \ge c_{k-1}^{(\alpha )}$. Furthermore,

$$\begin{aligned} c_k^{(\alpha )} - c_{k-1}^{(\alpha )} = (2-\alpha ) k^{1-\alpha }-2 k^{2-\alpha }+3(k-1)^{2-\alpha }-(k-2)^{2-\alpha }. \end{aligned}$$

Let $g(x) = (2-x) k^{1-x}-2 k^{2-x}+3(k-1)^{2-x}-(k-2)^{2-x}$, then for $x\in (0,1)$, $g'(x) >0$. Hence, g is monotone increasing and attains its maximum at $x=1$. Consequently, $g(x) \le g(1) = 0$ and the third part is complete. To complete the proof, it suffices to use the third part together with the showing that $c_1^{(\alpha )} \leqslant 2$. Suppose $g(x) = 2^{2-x} - 2$, then $g'(x) <0$ and g is monotone decreasing on (0, 1). Therefore $g(x) \leqslant g(0) =2$. $\square$

For constructing a finite-difference scheme, We utilize the difference operator L to approximate the diffusion term and the S2 discretization (A.1) for approximating the Caputo fractional derivative (1.2). Hence, problem (1.1) is approximated using the following discrete problem:

$$\begin{aligned}&D ^{\alpha , 2}_t u^k_i = Lu^k_i + f(x_i,t_k), \quad i=1,2,\ldots ,m, \quad k=1,2,\ldots ,n, \end{aligned}$$

(A.3)

$$\begin{aligned}&u_0^{k} = u_{m}^{k} = 0, \qquad k=0,1,\ldots ,n, \end{aligned}$$

(A.4)

where

$$\begin{aligned}&u_i^k = \sum _{j=0}^{k+1} \beta _{i}^{j} B_{j}(t^k) = \frac{1}{2} (\beta _{i}^{k} + \beta _{i}^{k+1}), \qquad i=0,\ldots ,m, \quad k=0,\ldots ,n, \end{aligned}$$

(A.5)

$$\begin{aligned}&\partial _t u(x_i, 0) \simeq \sum _{j=0}^{1} \beta _{i}^{j} \partial _t B_{j}(0) = \dfrac{1}{\Delta t} ( \beta _{i}^{1} - \beta _{i}^{0}), \quad i=0,\ldots , m. \end{aligned}$$

(A.6)

1.1 Stability and Convergence Analyses

Theorem A.1

Difference scheme (A.3) is unconditionally stable and its solution satisfies

$$\begin{aligned} \Vert u^k \Vert \leqslant \Vert u^0 \Vert , \qquad k = 1, 2, \ldots , n. \end{aligned}$$

Proof

Taking the inner product of (A.3) with $u^k$, and analysis similar to that in the proof of Theorem 3.1 leads to the following cases.

Case 1: If $c_0^{(\alpha )} \ge c_1^{(\alpha )}$, then for $\varepsilon \ge 0$, we get $\Vert \beta ^{k+1} \Vert ^2\, \leqslant \varepsilon \, \Vert \beta ^{0}\Vert ^2+ \sum _{j=0}^{k} w_j \,\Vert \beta ^{j}\Vert ^2,$ where

$$\begin{aligned} w_j = \left\{ \begin{array}{ll} 4 c_k^{(\alpha )}, &{} j=0,\\ 4 (c_{0}^{(\alpha )} - c_{1}^{(\alpha )}) +6 , &{} j = k, \\ 4(c_{k-j}^{(\alpha )} - c_{k-j+1}^{(\alpha )}) , &{} \text {otherwise}. \end{array} \right. \end{aligned}$$

According to Lemma A.1, $w_j$’s are nonnegative and using Lemma 2.4, we have

$$\begin{aligned} \Vert \beta ^{k+1} \Vert ^2\, \leqslant \varepsilon \exp \Big (\sum _{j=0}^{k} w_j\Big ) \, \Vert \beta ^{0}\Vert ^2=\varepsilon \exp (C) \, \Vert \beta ^{0}\Vert ^2, \end{aligned}$$

with $C = 2(2 c_{0}^{(\alpha )} + 3)>0$. Choosing $\varepsilon \le 1/\exp (C)$ gives $\Vert \beta ^{k+1}\Vert \leqslant \Vert \beta ^{0}\Vert$ for $k = 1, \ldots , n$. Similarly to proof of Theorem 3.1, it leads to the unconditional stability of difference scheme (A.3) in the maximum- and discrete $L^2$-norm.

Case 2: If $c_0^{(\alpha )} < c_1^{(\alpha )}$, then

$$\begin{aligned} \frac{1}{2}\Vert \beta ^{k+1} \Vert ^2&\leqslant (1-\frac{1}{2} c^{ (\alpha )}_1)\Vert \beta ^{k+1} \Vert ^2 \leqslant 4 \sum _{j=1}^{k-1} (c^{ (\alpha )}_{k-j} - c^{ (\alpha )}_{k-j+1})\, \Vert \beta ^{j} \Vert ^2 + 4 c_{k}^{(\alpha )} \Vert \beta ^{0} \Vert ^2 + \frac{17 }{8} c^{ (\alpha )}_{1} \Vert \beta ^{k} \Vert ^2,\nonumber \\&\leqslant 4 \sum _{j=1}^{k-1} (c^{ (\alpha )}_{k-j} - c^{ (\alpha )}_{k-j+1})\, \Vert \beta ^{j} \Vert ^2 + 4 c_{k}^{(\alpha )} \Vert \beta ^{0} \Vert ^2 + 5 \Vert \beta ^{k} \Vert ^2, \end{aligned}$$

(A.7)

where Lemma A.1 was used. For $\varepsilon \ge 0$, we now get $\Vert \beta ^{k+1} \Vert ^2\, \leqslant \varepsilon \, \Vert \beta ^{0}\Vert ^2+ \sum _{j=0}^{k} w_j \,\Vert \beta ^{j}\Vert ^2,$ where

$$\begin{aligned} w_j = 2\left\{ \begin{array}{l@{\qquad }l} 4 c_k^{(\alpha )}, &{} j=0,\\ 5, &{} j = k, \\ 4 (c_{k-j}^{(\alpha )} - c_{k-j+1}^{(\alpha )}) , &{} \text {otherwise}. \end{array} \right. \end{aligned}$$

According to Lemma A.1, $w_j$’s are nonnegative and using Lemma 2.4, we have

$$\begin{aligned} \Vert \beta ^{k+1} \Vert ^2\, \leqslant \varepsilon \exp \Big (\sum _{j=0}^{k} w_j\Big ) \, \Vert \beta ^{0}\Vert ^2=\varepsilon \exp (C) \, \Vert \beta ^{0}\Vert ^2, \end{aligned}$$

where $C = 2 (4 c_{1}^{(\alpha )} + 5)$. Obviously, $C>0$ and choosing $\varepsilon \le 1/\exp (C)$ gives $\Vert \beta ^{k+1}\Vert \leqslant \Vert \beta ^{0}\Vert$ for $k = 1, \ldots , n$. Then the same way as in Case 1 applies to show that the difference scheme (A.3) is unconditionally stable in the maximum- and discrete $L^2$-norm in this Case. $\square$

Now, we consider the local truncation error and define

$$\begin{aligned} \tau (x,t)&:= D^{\alpha , 2}_t {u}(x,s) - \frac{1}{h^2} \big (b(x+h/2,t) {u}(x+h,t) - (b(x-h/2,t) \nonumber \\&\quad + b(x+h/2,t)) {u}(x,t)+ b(x-h/2,t) {u}(x-h,t)\big ) - f(x,t), \end{aligned}$$

(A.8)

to establish an error estimate for difference scheme (A.3). Let $e_i^k =u(x_i,t_k) - u_i^k$, where $u(x_i,t_k)$ and $u_i^k$ are the solutions of problem (1.1) at grid point $(x_i,t_k)$ and the difference scheme (A.3), repectively. Subtracting Eq. (A.3) from Eq. (A.8) for $x=x_i$ and $t=t_k$ leads to

$$\begin{aligned} \tau _i^k= D^{\alpha , 2}_t e_i^k - L e_i^k. \end{aligned}$$

(A.9)

Obviously, boundary conditions imply $e_0^k = e_m^k= 0$ for $k=1,\ldots ,n$ which leads to

$$\begin{aligned} D^{\alpha , 2}_t e_i^k = \dfrac{\Delta t^{-\alpha }}{\Gamma (3-\alpha )} \left( {\mathcal {E}}^{k+1}_{i} - \sum _{j=2}^{k} (c^{ (\alpha )}_{k-j} - c^{ (\alpha )}_{k-j+1}) \, {\mathcal {E}}^{j}_{i} - (c_{k-1}^{(\alpha )} - 2c_{k}^{(\alpha )}) \, {\mathcal {E}}^{1}_i \right) . \end{aligned}$$

Theorem A.2

[28] If $v \in \mathrm{{C}}^3[0, t_k]$ and $R_2(v(t_k)) = {\mathcal {D}}^{\alpha }_t v(t)\big \vert _{t = t_k} - D ^{\alpha ,2}_t v(t)\big \vert _{t = t_k}$ then

$$\begin{aligned} \vert R_2(v(t_k)) \vert \leqslant \dfrac{M_{ttt}}{2} \Delta t^{3 - \alpha }, \qquad M_{ttt} = \max _{0 \leqslant t \leqslant t_k} |v^{(3)}(t)|. \end{aligned}$$

Theorem A.3

If $u\in \mathrm{{C}}^{4,3}(\Omega \times I)$, difference scheme (A.3) is consistent with $3-\alpha$ order accuracy.

Proof

The proof is the same way as Theorem 3.4. $\square$

Theorem A.4

Under the condition of Theorem A.3, difference scheme (A.3) is convergent.

Proof

Taking the inner product of (A.9) with $e^k$, and using the same way as in the proof of Theorem 3.1 leads to the following cases.

Case 1: If $c_{0}^{(\alpha )} \ge c_{1}^{(\alpha )}$, then $\Vert {\mathcal {E}}^{k+1} \Vert ^2 \leqslant \dfrac{4}{ c_{k}^{(\alpha )}} \mu ^2 \Vert \mathcal {\tau }^{k} \Vert ^2 + \sum _{j=1}^{k}w_j \Vert {\mathcal {E}}^{j} \Vert ^2,$ where

$$\begin{aligned} w_j =\dfrac{1}{c_{k}^{(\alpha )}}\left\{ \begin{array}{l@{\qquad }l} 4 c_{k-1}^{(\alpha )}, &{} j = 1, \\ 6 - 4 c_{1}^{(\alpha )}, &{} j = k, \\ 4\left( c_{k-j}^{(\alpha )} - c_{k-j+1}^{(\alpha )} \right) , &{} \text {otherwise}. \end{array} \right. \end{aligned}$$

Using Lemma 2.4, we have $\Vert {\mathcal {E}}^{k+1} \Vert \leqslant \frac{2 \mu }{\sqrt{c_{k}^{(\alpha )}}}\Vert \tau ^k \Vert \exp (3) \leqslant 2 \mu \exp (3) \Vert \tau ^k \Vert$ and utilizing Theorem A.3, complete the proof.

Case 2: If $c_{0}^{(\alpha )} < c_{1}^{(\alpha )}$, then $\Vert {\mathcal {E}}^{k+1} \Vert ^2 \leqslant 8 \mu ^2 \Vert \mathcal {\tau }^{k} \Vert ^2 + \sum _{j=1}^{k}w_j \Vert {\mathcal {E}}^{j} \Vert ^2,$ where

$$\begin{aligned} w_j = 4 \left\{ \begin{array}{l@{\qquad }l} c_{k-1}^{(\alpha )}, &{} j = 1, \\ 2, &{} j = k, \\ (c_{k-j}^{(\alpha )} - c_{k-j+1}^{(\alpha )}), &{} \text {otherwise}. \end{array} \right. \end{aligned}$$

Using Lemma 2.4, we have

$$\begin{aligned} \Vert {\mathcal {E}}^{k+1} \Vert \leqslant \sqrt{8} \mu \exp (C_1) \Vert \tau ^k \Vert , \end{aligned}$$

(A.10)

where $C_1 = (4 + 2 c_{1}^{(\alpha )})$. Using Theorem A.3, $\Vert {\mathcal {E}}^k \Vert \leqslant C\Delta t^{3-\alpha }$ for $k=1, 2, \ldots , n$. Eventually $\Vert e^k \Vert \leqslant C\Delta t^{3-\alpha }$ and due to Lemma 2.2, the scheme has the convergence order ${\mathcal {O}}(3 - \alpha )$ in the maximum- and $L^2$-norm. $\square$