Fast High-Order Compact Finite Difference Methods Based on the Averaged L1 Formula for a Time-Fractional Mobile-Immobile Diffusion Problem

Abstract

A two-dimensional time-fractional mobile-immobile diffusion problem with the Caputo time-fractional derivative of order $\alpha \in (0,1)$ is considered. We show that the solution of the problem has a weak singularity at the initial time. Using the averaged L1 formula to approximate the Caputo time-fractional derivative and using a compact finite difference approximation to discretize the space derivatives, we propose a high-order averaged L1-type compact finite difference method on the uniform space-time mesh for the problem. We then base on this method to develop an averaged L1-type compact alternating direction implicit (ADI) finite difference method and a fast sum-of-exponentials compact ADI finite difference method, both of which significantly reduce the storage requirements and the computational costs while maintaining the same global convergence rate. By using the discrete energy analysis technique, we rigorously prove that all methods are unconditionally stable and convergent, and they have the spatial global fourth-order convergence rate and the temporal global convergence rate of order $\min \{2, 3-2\alpha \}$. For the case of $\alpha >1/2$, we use the discrete minimum-maximum principle to prove that the temporal second-order convergence rate can also be achieved in positive time. Numerical results confirm the theoretical analysis results and demonstrate the computational efficiency of the methods.

A Appendix: Proof of Theorem 11

Proof

When the spatial mesh is isotropic, that is, $h_{x}=h_{y}=h$, the operators ${{\mathcal {H}}}$ and ${{\mathcal {Q}}}$ can be written as

$$\begin{aligned} {{\mathcal {H}}}z_{i,j}=\sum _{k_{1}, k_{2}=-1}^{1} p_{i,j}^{(k_{1}, k_{2})} z_{i+k_{1}, j+k_{2}},\qquad {{\mathcal {Q}}}z_{i,j}=\frac{1}{h^{2}} \sum _{k_{1}, k_{2}=-1}^{1} q_{i,j}^{(k_{1}, k_{2})} z_{i+k_{1}, j+k_{2}}, \end{aligned}$$

(118)

where

$$\begin{aligned}{} & {} p_{i,j}^{(\pm 1,\pm 1)}=\frac{1}{144}, \qquad p_{i,j}^{(\pm 1,0)}=p_{i,j}^{(0,\pm 1)}=\frac{5}{72}, \qquad p_{i,j}^{(0,0)}=\frac{25}{36},\\{} & {} q_{i,j}^{(\pm 1,\pm 1)}=\frac{1}{6},\qquad \quad q_{i,j}^{(\pm 1,0)}=q_{i,j}^{(0,\pm 1)}=\frac{2}{3}, \qquad ~~ q_{i,j}^{(0,0)}=-\frac{10}{3}. \end{aligned}$$

Then we can write (58a) in the form:

$$\begin{aligned}{} & {} \left( \frac{\omega _{1,1}^{*}}{\tau }p_{i,j}^{(0,0)}- \frac{\kappa }{2h^{2}} q_{i,j}^{(0,0)}\right) u_{i,j}^{1}=\sum _{{k_{1}, k_{2}=-1}\begin{array}{c} (k_{1},k_{2})\not =(0,0) \end{array}}^{1}\left( \frac{\kappa }{2h^{2}}q_{i,j}^{(k_{1},k_{2})}- \frac{\omega _{1,1}^{*}}{\tau }p_{i,j}^{(k_{1},k_{2})} \right) u_{i+k_{1}, j+k_{2}}^{1}\nonumber \\{} & {} ~~~~~~~~~~~~+\sum _{{k_{1}, k_{2}=-1}}^{1}\left( \frac{\omega _{1,1}^{*}}{\tau }p_{i,j}^{(k_{1},k_{2})}+ \frac{\kappa }{2h^{2}}q_{i,j}^{(k_{1},k_{2})} \right) u_{i+k_{1}, j+k_{2}}^{0}+{{\mathcal {H}}}g_{i,j}^{1}, \qquad (i,j)\in \Omega _{h}, \end{aligned}$$

(119)

and for $2\le n\le N$,

$$\begin{aligned}{} & {} \left( \frac{\omega _{n,n}^{*}}{\tau }p_{i,j}^{(0,0)}- \frac{\kappa }{2h^{2}} q_{i,j}^{(0,0)}\right) u_{i,j}^{n}=\sum _{{k_{1}, k_{2}=-1}\begin{array}{c} (k_{1},k_{2}) \not =(0,0) \end{array}}^{1}\left( \frac{\kappa }{2h^{2}}q_{i,j}^{(k_{1},k_{2})}- \frac{\omega _{n,n}^{*}}{\tau }p_{i,j}^{(k_{1},k_{2})} \right) u_{i+k_{1}, j+k_{2}}^{n}\nonumber \\{} & {} ~~~~+\sum _{{k_{1}, k_{2}=-1}}^{1}\left( \frac{\omega _{n,n}^{*} -\omega _{n,n-1}^{*}}{\tau }p_{i,j}^{(k_{1},k_{2})}+ \frac{\kappa }{2h^{2}}q_{i,j}^{(k_{1},k_{2})} \right) u_{i+k_{1}, j+k_{2}}^{n-1}+\zeta _{i,j}^{n}, \qquad (i,j)\in \Omega _{h},\nonumber \\ \end{aligned}$$

(120)

where

$$\begin{aligned}{} & {} \zeta _{i,j}^{n}=\frac{1}{\tau } \sum _{k=1}^{n-2} \left( \omega _{n,k+1}^{*} -\omega _{n,k}^{*}\right) \sum _{{k_{1}, k_{2}=-1}}^{1}p_{i,j}^{(k_{1},k_{2})} u_{i+k_{1}, j+k_{2}}^{k}\nonumber \\{} & {} ~~~~~~~~~~~~~~~~+\frac{1}{\tau } \omega _{n,1}^{*} \sum _{{k_{1}, k_{2}=-1}}^{1}p_{i,j}^{(k_{1},k_{2})} u_{i+k_{1}, j+k_{2}}^{0}+\mathcal{H}g_{i,j}^{n}. \end{aligned}$$

(121)

When the mesh conditions (98) and (99) are satisfied, $\omega _{n,k+1}^{*}> \omega _{n,k}^{*}$ for $1\le k\le n-1$ (see (40)) and all coefficients of the terms on the right-hand side of (119) and (120) are nonnegative. We also observe that $\sum _{k_{1}, k_{2}=-1}^{1}p_{i,j}^{(k_{1},k_{2})}=1$ and $\sum _{k_{1}, k_{2}=-1}^{1}q_{i,j}^{(k_{1},k_{2})}=0$ for all $(i,j)\in \Omega _{h}$.

Now suppose that $u_{i,j}^{1}$ attains its maximum in $\overline{\Omega }_{h}$ at $(i_{0}, j_{0})\in \Omega _{h}$. Then we have from (119) that

$$\begin{aligned} \left( \frac{\omega _{1,1}^{*}}{\tau }p_{i_{0},j_{0}}^{(0,0)}- \frac{\kappa }{2h^{2}} q_{i_{0},j_{0}}^{(0,0)}\right) u_{i_{0},j_{0}}^{1}\le & {} \left( \frac{\omega _{1,1}^{*}}{\tau }p_{i_{0},j_{0}}^{(0,0)}- \frac{\kappa }{2h^{2}} q_{i_{0},j_{0}}^{(0,0)}-\frac{1}{\tau } \omega _{1,1}^{*}\right) u_{i_{0},j_{0}}^{1}\nonumber \\{} & {} +\frac{1}{\tau } \omega _{1,1}^{*} \Theta _{\max }^{1}.~~~~~~ \end{aligned}$$

(122)

This shows $u_{i_{0},j_{0}}^{1}\le \Theta _{\max }^{1}$. Since $u_{i,j}^{1}=0$ for all $(i,j)\in \partial \Omega _{h}$, we obtain $u_{i,j}^{1}\le \Theta _{\max }^{1}$ for all $(i,j)\in \overline{\Omega }_{h}$. An identical argument shows $u_{i,j}^{1}\ge \Theta _{\min }^{1}$ for all $(i,j)\in \overline{\Omega }_{h}$. Assume, by induction, that $\Theta _{\min }^{n}\le u_{i,j}^{n}\le \Theta _{\max }^{n}$ for all $(i,j)\in \overline{\Omega }_{h}$ and $1\le n\le n_{0}-1$ $(n_{0}\ge 2)$. Suppose that $u_{i,j}^{n_{0}}$ attains its maximum in $\overline{\Omega }_{h}$ at $(i_{0}, j_{0})\in \Omega _{h}$. Then we use (120) and the inductive hypothesis to get a similar inequality (122), where subscript or superscript 1 is replaced by $n_{0}$. By this inequality, we deduce $u_{i_{0},j_{0}}^{n_{0}}\le \Theta _{\max }^{n_{0}}$ which implies $u_{i,j}^{n_{0}}\le \Theta _{\max }^{n_{0}}$ for all $(i,j)\in \overline{\Omega }_{h}$ because $u_{i,j}^{n_{0}}=0$ for all $(i,j)\in \partial \Omega _{h}$. The same argument can be applied to prove $u_{i,j}^{n_{0}}\ge \Theta _{\min }^{n_{0}}$ for all $(i,j)\in \overline{\Omega }_{h}$. The principle of mathematical induction completes the proof. $\square$