A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection–diffusion equations

Abstract

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.

Appendix

In this appendix, we prove Propositions 5.1–5.3.

Proof of Proposition 5.1

When $\frac{\partial u}{\partial t}(x,t)$ is bounded, $\frac{\partial ^{\alpha } u}{\partial t^{\alpha }}(x,0)=0$ (see [8]). Also, we have $\frac{\partial ^{2} u}{\partial x^{2}}(x,0)=0$ since $u(x,0)=0$ for all $x\in [0,L]$. Thus, by the governing equation of (2.1),

$$\begin{aligned} \displaystyle \frac{\partial u}{\partial t}(x,0)=-\beta \frac{\partial ^{\alpha } u}{\partial t^{\alpha }}(x,0)+D \displaystyle \frac{\partial ^{2} u}{\partial x^{2}}(x,0)+q(x) u(x,0)+g(x,0)=g(x,0). \end{aligned}$$

This proves (5.19). $\square$

Proof of of Proposition 5.2

By integrating by parts and by (5.19),

$$\begin{aligned} \frac{\partial ^{\alpha } u}{\partial t^{\alpha }}(x,t)= \frac{g(x,0)}{{\varGamma }(2-\alpha )} t^{1-\alpha }+\frac{1}{{\varGamma }(2-\alpha )} \int _{0}^{t} \frac{\partial ^{2} u}{\partial s^{2}}(x,s) (t-s)^{1-\alpha }\mathrm{d}s. \end{aligned}$$

This implies that the governing equation of (2.1) can be written as

$$\begin{aligned}&\frac{\partial u}{\partial t}(x,t)+\frac{\beta }{{\varGamma }(2-\alpha )} \int _{0}^{t} \frac{\partial ^{2} u}{\partial s^{2}}(x,s) (t-s)^{1-\alpha } \mathrm{d}s = D \frac{\partial ^{2} u}{\partial x^{2}}(x,t) \\&\quad +\,q(x) u(x,t)+F(x,t). \end{aligned}$$

Since $u(x,t)\in \mathcal{C}^{2,2}([0,L]\times [0,T])$, we have

$$\begin{aligned} \frac{\partial F}{\partial t}(x,t)= & {} \frac{\partial ^{2} u}{\partial t^{2}}(x,t)+\beta {_{~0}^{C}}\mathcal{D}_{t}^{1+\alpha } u(x,t)-D \frac{\partial ^{3} u}{\partial t\partial x^{2}}(x,t) \\&-\,q(x)\frac{\partial u}{\partial t}(x,t) \end{aligned}$$

and

$$\begin{aligned} \displaystyle \lim _{t\rightarrow 0} \frac{\partial F}{\partial t}(x,t)= & {} \frac{\partial ^{2} u}{\partial t^{2}}(x,0)+\beta {_{~0}^{C}}\mathcal{D}_{t}^{1+\alpha } u(x,0)-D \frac{\partial ^{3} u}{\partial t\partial x^{2}}(x,0) -q(x)\frac{\partial u}{\partial t}(x,0)\\= & {} \displaystyle \frac{\partial ^{2} u}{\partial t^{2}}(x,0)-D \frac{\partial ^{2} ~}{\partial x^{2}} \left( \frac{\partial u}{\partial t}(x,0)\right) -q(x)\frac{\partial u}{\partial t}(x,0). \end{aligned}$$

This together with (5.19) proves (5.21). $\square$

Proof of of Proposition 5.3

By integrating by parts twice and by (5.19),

$$\begin{aligned} \frac{\partial ^{\alpha } u}{\partial t^{\alpha }}(x,t)= & {} \frac{g(x,0)}{{\varGamma }(2-\alpha )} t^{1-\alpha }+ \frac{1}{{\varGamma }(3-\alpha )} \frac{\partial ^{2} u}{\partial t^{2}}(x,0) t^{2-\alpha }\\&+ \frac{1}{{\varGamma }(3-\alpha )} \int _{0}^{t} \frac{\partial ^{3} u}{\partial s^{3}}(x,s)(t-s)^{2-\alpha }\mathrm{d}s. \end{aligned}$$

We therefore have from the governing equation of (2.1) that

$$\begin{aligned}&\frac{\partial u}{\partial t}(x,t)+\frac{\beta }{{\varGamma }(3-\alpha )} \int _{0}^{t} \frac{\partial ^{3} u}{\partial s^{3}}(x,s) (t-s)^{2-\alpha } \mathrm{d}s\\&\quad = D \frac{\partial ^{2} u}{\partial x^{2}}(x,t) +q(x) u(x,t)+G(x,t). \end{aligned}$$

Since $u(x,t)\in \mathcal{C}^{2,3}([0,L]\times [0,T])$, we have

$$\begin{aligned} \frac{\partial ^{2} G}{\partial t^{2}}(x,t)=\frac{\partial ^{3} u}{\partial t^{3}}(x,t)+\beta {_{~0}^{C}}{D}_{t}^{2+\alpha } u(x,t)-D \frac{\partial ^{4} u}{\partial t^{2}\partial x^{2}}(x,t) -q(x)\frac{\partial ^{2} u}{\partial t^{2}}(x,t) \end{aligned}$$

and

$$\begin{aligned} \displaystyle \lim _{t\rightarrow 0} \frac{\partial ^{2} G}{\partial t^{2}}(x,t)= & {} \frac{\partial ^{3} u}{\partial t^{3}}(x,0)+\beta {_{~0}^{C}}\mathcal{D}_{t}^{2+\alpha } u(x,0)-D \frac{\partial ^{4} u}{\partial t^{2}\partial x^{2}}(x,0) -q(x)\frac{\partial ^{2} u}{\partial t^{2}}(x,0)\\= & {} \displaystyle \frac{\partial ^{3} u}{\partial t^{3}}(x,0)-D \frac{\partial ^{2} ~}{\partial x^{2}} \left( \frac{\partial ^{2} u}{\partial t^{2}}(x,0)\right) -q(x)\frac{\partial ^{2} u}{\partial t^{2}}(x,0). \end{aligned}$$

This proves that the limit $\lim \nolimits _{t\rightarrow 0} \frac{\partial ^{2} G}{\partial t^{2}}(x,t)$ exists and (5.23) holds true. $\square$