High-Order Compact Difference Methods for Caputo-Type Variable Coefficient Fractional Sub-diffusion Equations in Conservative Form

Abstract

A set of high-order compact finite difference methods is proposed for solving a class of Caputo-type fractional sub-diffusion equations in conservative form. The diffusion coefficient of the equation may be spatially variable, and the proposed methods have the global convergence order $\mathcal{O}(\tau ^{r}+h^{4})$, where $r\ge 2$ is a positive integer and $\tau$ and h are the temporal and spatial steps. Such new high-order compact difference methods greatly improve the known methods in the literature. The local truncation error and the solvability of the methods are discussed in detail. By applying a discrete energy technique to the matrix form of the methods, a rigorous theoretical analysis of the stability and convergence of the methods is carried out for the case of $2\le r\le 6$, and the optimal error estimates in the weighted $H^{1}$, $L^{2}$ and $L^{\infty }$ norms are obtained for the general case of variable coefficient. Applications are given to two model problems, and some numerical results are presented to illustrate the various convergence orders of the methods.

Appendix

Proof of Proposition 5.1

“$\Longrightarrow$”: It follows from the result (1) of Lemma 2.2 with $r_{0}=r$.

“$\Longleftarrow$”: Let $y_\mathrm{ex}(t)$ be the extension defined by (2.9) (with r replaced by $r+1$). Since $y^{(k)}(0)=0$ for $k=0,1,\dots ,r$, we have from the proof of the result (1) of Lemma 2.2 (with $r_{0}=r$ and $r^{\prime }=r+1$) that

$$\begin{aligned} \lim _{|\omega |\rightarrow \infty } \left| \omega \right| ^{2-\alpha } \left| \omega \right| ^{r+\alpha } \left| \hat{y}_\mathrm{ex}(\omega ) \right| = \lim _{|\omega |\rightarrow \infty } \left| \omega \right| ^{r+2} \left| \hat{y}_\mathrm{ex}(\omega ) \right| =\left| y^{(r+1)}(0)\right| . \end{aligned}$$

This implies that the function $|\omega |^{r+\alpha } \left| \hat{y}_\mathrm{ex}(\omega ) \right|$ is integrable on $\mathbb {R}$ because of $2-\alpha >1$, and so $y_\mathrm{ex}(t)\in \mathscr {C}^{r+\alpha }(\mathbb {R})$, i.e., $y(t)\in \mathscr {C}^{r+\alpha }[0,T]$. $\square$

Proof of Proposition 5.2

Since $u(x,t)\in C^{0,1}([0,L]\times [0,T])$ and $\mathcal{L}u(x,t)\in C^{0,1}([0,L]\times [0,T])$, we apply the Caputo fractional derivative operator ${_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha }$ to the governing equation of (1.1) and make use of Lemma 3.13 in [59] to obtain

$$\begin{aligned} \partial _{t} u(x,t)={_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha }(\mathcal{L}u)(x,t)+{_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha }f(x,t), \qquad x\in [0,L]. \end{aligned}$$

(A.1)

By Lemma 3.11 in [59], ${_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha }( \mathcal{L}u )(x,0)=0$. This proves (5.3). $\square$

Proof of Proposition 5.3

By integrating by parts, we have that for $p=1,2,\dots ,r-1$,

$$\begin{aligned}&{_{~0}^{C}}\mathcal{D}_{t}^{\alpha }u(x,t)=\frac{1}{\Gamma (1-\alpha )} \int _{0}^{t} \partial _{s}u(x,s) (t-s)^{-\alpha } \mathrm{d} s\nonumber \\&\quad = \frac{t^{1-\alpha }}{\Gamma (2-\alpha )} \partial _{s} u(x,0)+\frac{1}{\Gamma (2-\alpha )} \int _{0}^{t} \partial _{s}^{2}u(x,s) (t-s)^{1-\alpha } \mathrm{d} s\nonumber \\&\quad = \sum _{l=1}^{p} \frac{t^{l-\alpha }}{\Gamma (l+1-\alpha )} \partial _{s}^{l} u(x,0)+\frac{1}{\Gamma (p+1-\alpha )} \int _{0}^{t} \partial _{s}^{p+1}u(x,s) (t-s)^{p-\alpha } \mathrm{d} s.~~~~\nonumber \\ \end{aligned}$$

(A.2)

This shows that for $p=1,2,\dots ,r-1$,

$$\begin{aligned} \partial _{t}^{p} ({_{~0}^{C}}\mathcal{D}_{t}^{\alpha })u(x,t)=\sum _{l=1}^{p} \frac{t^{l-\alpha -p}}{\Gamma (l-\alpha -p+1)} \partial _{s}^{l} u(x,0)+{_{~0}^{C}}\mathcal{D}_{t}^{\alpha }(\partial _{t}^{p}u)(x,t)~~(t\not =0). \nonumber \\ \end{aligned}$$

(A.3)

Similarly, for $p=2,3,\dots ,r$,

$$\begin{aligned} \partial _{t}^{p-1} ({_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha })(\mathcal{L}u)(x,t)= & {} \sum _{l=1}^{p-1} \frac{t^{l+\alpha -p}}{\Gamma (l+\alpha -p+1)} \partial _{s}^{l} (\mathcal{L}u)(x,0)\nonumber \\&+\,{_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha }(\partial _{t}^{p-1}(\mathcal{L}u))(x,t)~~(t\not =0). \end{aligned}$$

(A.4)

Differentiating the governing equation of (1.1) p times with respect to t and then solving for $\partial _{t}^{p}(\mathcal{L}u)(x,t)$, we obtain

$$\begin{aligned} \partial _{t}^{p}(\mathcal{L}u)(x,t)=\partial _{t}^{p}({_{~0}^{C}}\mathcal{D}_{t}^{\alpha })u(x,t)-\partial _{t}^{p}f(x,t)~~(t\not =0), \qquad p=1,2,\dots , r-1.\nonumber \\ \end{aligned}$$

(A.5)

This implies that for $x\in [0,L]$,

$$\begin{aligned} \partial _{t}^{p}(\mathcal{L}u)(x,0)=\lim _{t\rightarrow 0} \left( \partial _{t}^{p}({_{~0}^{C}}\mathcal{D}_{t}^{\alpha })u(x,t)-\partial _{t}^{p}f(x,t)\right) , \qquad p=1,2,\dots , r-1.\nonumber \\ \end{aligned}$$

(A.6)

Differentiating the Eq. (A.1) $p-1$ times with respect to t yields

$$\begin{aligned} \partial _{t}^{p} u(x,t)= & {} \partial _{t}^{p-1} ({_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha })(\mathcal{L}u)(x,t)+\partial _{t}^{p-1}({_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha }f)(x,t)~(t\not =0),\nonumber \\&p=2,3,\dots ,r, \end{aligned}$$

(A.7)

and so for $x\in [0,L]$,

$$\begin{aligned} \partial _{t}^{p} u(x,0)= & {} \lim _{t\rightarrow 0} \left( \partial _{t}^{p-1} ({_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha })(\mathcal{L}u)(x,t)+\partial _{t}^{p-1}({_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha }f)(x,t)\right) ,\nonumber \\&p=2,3,\dots ,r.~~~~ \end{aligned}$$

(A.8)

By Lemma 3.11 in [59], ${_{~0}^{C}}\mathcal{D}_{t}^{\alpha }(\partial _{t}^{p}u)(x,0)=0$ for $p=1,2,\dots ,r-1$ and ${_{~0}^{C}}\mathcal{D}_{t}^{1-\alpha }(\partial _{t}^{p-1}(\mathcal{L}u))(x,0)=0$ for $p=2,3,\dots ,r$. Then the result (5.4) follows by (A.3) and (A.6), and the result (5.5) follows by (A.4) and (A.8). The proof is completed. $\square$