On banded M-splitting iteration methods for solving discretized spatial fractional diffusion equations

Abstract

For solving time-dependent one-dimensional spatial-fractional diffusion equations of variable coefficients, we establish a banded M-splitting iteration method applicable to compute approximate solutions for the corresponding discrete linear systems resulting from certain finite difference schemes at every temporal level, and demonstrate its asymptotic convergence without imposing any extra condition. Also, we provide a multistep variant for the banded M-splitting iteration method, and prove that the computed solutions of the discrete linear systems by employing this iteration method converge to the exact solutions of the spatial fractional diffusion equations. Numerical experiments show the accuracy and efficiency of the multistep banded M-splitting iteration method.

7 Appendix: Proof of Theorem 4.1

In fact, it holds that

$$\begin{aligned} \Vert u^{\tau ,k_o}-u_{\star }^{\tau }\Vert _{\infty } \le \Vert u^{\tau ,k_o}-u^{\tau }\Vert _{\infty }+\Vert u^{\tau }-u_{\star }^{\tau }\Vert _{\infty }. \end{aligned}$$

(7.1)

As $u_{\star }^{\tau }$ is the exact solution of the fractional diffusion equation (1.1), from the approximation property of the finite-difference formulas it satisfies

$$\begin{aligned} A^{\tau }u_{\star }^{\tau }&=\triangle {t} \, f^{\tau }+u_{\star }^{\tau -1} \\&\quad -\frac{\triangle {t}^2}{2} \, \frac{\partial ^2 u_{\star }^{\tau }}{\partial {t^2}} -\left( 1-\frac{\alpha }{2}\right) h^{\alpha +1} \left( D^{\tau } \, \frac{\partial ^{\alpha +1} u_{\star }^{\tau }}{\partial _{+}{x^{\alpha +1}}} +W^{\tau } \, \frac{\partial ^{\alpha +1} u_{\star }^{\tau }}{\partial _{-}{x^{\alpha +1}}} \right) \\&\quad +\mathscr {O}(\triangle {t}^3)+\mathscr {O}(h^2 \, \triangle {t}). \end{aligned}$$

Hence, by making use of the discrete linear system (2.3) we can obtain the truncated error equation

$$\begin{aligned} A^{\tau }(u^{\tau }-u_{\star }^{\tau }) =u^{\tau -1}-u_{\star }^{\tau -1} +r^{\tau }(x,t) +\mathscr {O}(\triangle {t}^3)+\mathscr {O}(h^2 \, \triangle {t}), \end{aligned}$$

(7.2)

where

$$\begin{aligned} r^{\tau }(x,t) :=\frac{\triangle {t}^2}{2} \, \frac{\partial ^2 u_{\star }^{\tau }}{\partial {t^2}} +\left( 1-\frac{\alpha }{2}\right) h^{\alpha +1} \left( D^{\tau } \, \frac{\partial ^{\alpha +1} u_{\star }^{\tau }}{\partial _{+}{x^{\alpha +1}}} +W^{\tau } \, \frac{\partial ^{\alpha +1} u_{\star }^{\tau }}{\partial _{-}{x^{\alpha +1}}} \right) \end{aligned}$$

is the corresponding error term. With applications of the Lipschitz conditions in (4.2) and the definitions of the diagonal matrices $D^{\tau }$ and $W^{\tau }$ in (2.4), it can be verified straightforwardly that this error term satisfies

$$\begin{aligned} \Vert r^{\tau }(x,t)\Vert _{\infty } \le \varepsilon \, \triangle {t}. \end{aligned}$$

The error equation (7.2), together with the upper bound about $\Vert (A^{\tau })^{-1}\Vert _{\infty }$ in Lemma 2.1, straightforwardly lead to the estimate

$$\begin{aligned} \Vert u_{\star }^{\tau }-u^{\tau }\Vert _{\infty }&\le \Vert (A^{\tau })^{-1}\Vert _{\infty } \left( \Vert u^{\tau -1}-u_{\star }^{\tau -1}\Vert _{\infty } +\varepsilon \, \triangle {t} +\mathscr {O}(\triangle {t}^3)+\mathscr {O}(h^2 \, \triangle {t})\right) \\&\le \frac{1}{1+\eta _{\min } \, \triangle {t}} \left( \Vert u^{\tau -1}-u_{\star }^{\tau -1}\Vert _{\infty } +\varepsilon \, \triangle {t} +\mathscr {O}(\triangle {t}^3)+\mathscr {O}(h^2 \, \triangle {t})\right) \\&\le \frac{1}{(1+\eta _{\min } \, \triangle {t})^{\tau }} \Vert u^{0}-u_{\star }^{0}\Vert _{\infty } +\frac{1}{\eta _{\min } \, \triangle {t}} \left( \varepsilon \, \triangle {t} +\mathscr {O}(\triangle {t}^3)+\mathscr {O}(h^2 \, \triangle {t})\right) \\&=\frac{1}{(1+\eta _{\min } \, \triangle {t})^{\tau }} \Vert u^{0}-u_{\star }^{0}\Vert _{\infty } +\frac{1}{\eta _{\min }} \left( \varepsilon +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \right) . \end{aligned}$$

Note that $u^{0}=u_{\star }^{0}$. Then it holds that

$$\begin{aligned} \Vert u^{\tau }-u_{\star }^{\tau }\Vert _{\infty } \le \frac{\varepsilon }{\eta _{\min }} +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2). \end{aligned}$$

(7.3)

Moreover, by the Lipschitz continuity of the solution $u_{\star }(x,t)$ of the fractional diffusion equation (1.1), we can obtain

$$\begin{aligned} \Vert u^{\tau }-u^{\tau -1}\Vert _{\infty }&\le \Vert u_{\star }^{\tau }-u_{\star }^{\tau -1}\Vert _{\infty } +\Vert u^{\tau }-u_{\star }^{\tau }\Vert _{\infty } +\Vert u^{\tau -1}-u_{\star }^{\tau -1}\Vert _{\infty } \nonumber \\&\le \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2). \end{aligned}$$

(7.4)

On the other hand, based on the discrete linear system (2.3) and the MBMS iteration method (4.1), we can obtain the iterative error equation

$$\begin{aligned} u^{\tau ,k}-u^{\tau }&=\left[ (M^{\tau })^{-1}N^{\tau } u^{\tau ,k-1} +(M^{\tau })^{-1} \left( \triangle {t} \, f^{\tau }+u^{\tau -1,k_o} \right) \right] \\&\quad -\left[ (M^{\tau })^{-1}N^{\tau } u^{\tau } +(M^{\tau })^{-1} \left( \triangle {t} \, f^{\tau }+u^{\tau -1} \right) \right] \\&=(M^{\tau })^{-1}N^{\tau } \left( u^{\tau ,k-1}-u^{\tau } \right) +(M^{\tau })^{-1} \left( u^{\tau -1,k_o}-u^{\tau -1} \right) \\&= \cdots \\&=\left( (M^{\tau })^{-1}N^{\tau }\right) ^k (u^{\tau ,0}-u^{\tau }) +\sum \limits _{i=0}^{k-1}\left( (M^{\tau })^{-1}N^{\tau }\right) ^i (M^{\tau })^{-1} (u^{\tau -1,k_o}-u^{\tau -1}), \end{aligned}$$

which implies

$$\begin{aligned} \left| u^{\tau ,k_o}-u^{\tau }\right|\le & {} \left( (M^{\tau })^{-1}N^{\tau }\right) ^{k_o} \left| u^{\tau ,0}-u^{\tau }\right| +\sum \limits _{i=0}^{k_o-1}\left( (M^{\tau })^{-1}N^{\tau }\right) ^i (M^{\tau })^{-1} \left| u^{\tau -1,k_o}-u^{\tau -1}\right| \\\le & {} \left( (M^{\tau })^{-1}N^{\tau }\right) ^{k_o} \left| u^{\tau ,0}-u^{\tau }\right| +\sum \limits _{i=0}^{\infty }\left( (M^{\tau })^{-1}N^{\tau }\right) ^i (M^{\tau })^{-1} \left| u^{\tau -1,k_o}-u^{\tau -1}\right| \\= & {} \left( (M^{\tau })^{-1}N^{\tau }\right) ^{k_o} \left| u^{\tau ,0}-u^{\tau }\right| \\&+\sum \limits _{i=0}^{\infty }\left( (M^{\tau })^{-1}N^{\tau }\right) ^i \left[ (M^{\tau })^{-1}A^{\tau }\right] (A^{\tau })^{-1} \left| u^{\tau -1,k_o}-u^{\tau -1}\right| \\= & {} \left( (M^{\tau })^{-1}N^{\tau }\right) ^{k_o} \left| u^{\tau ,0}-u^{\tau }\right| +(A^{\tau })^{-1} \left| u^{\tau -1,k_o}-u^{\tau -1}\right| . \end{aligned}$$

Therefore, it follows from Lemma 2.1 and Theorem 3.1 that

$$\begin{aligned} \Vert u^{\tau ,k_o}-u^{\tau }\Vert _{\infty }\le & {} \Vert (M^{\tau })^{-1}N^{\tau }\Vert _{\infty }^{k_o} \, \Vert u^{\tau ,0}-u^{\tau }\Vert _{\infty } +\Vert (A^{\tau })^{-1}\Vert _{\infty } \, \Vert u^{\tau -1,k_o}-u^{\tau -1}\Vert _{\infty } \nonumber \\\le & {} \sigma ^{k_o} \Vert u^{\tau ,0}-u^{\tau }\Vert _{\infty } +\frac{1}{1+\eta _{\min } \, \triangle {t}} \Vert u^{\tau -1,k_o}-u^{\tau -1}\Vert _{\infty }. \end{aligned}$$

(7.5)

Note that $u^{\tau ,0}=2u^{\tau -1,k_o}-u^{\tau -2,k_o}$ if $\tau \ge 2$, and $u^{\tau ,0}=u^0$ if $\tau =1$. Based on the equivalent expressions

$$\begin{aligned} u^{\tau ,0}-u^{\tau } =2(u^{\tau -1,k_o}-u^{\tau -1}) -(u^{\tau -2,k_o}-u^{\tau -2}) -(u^{\tau }-u^{\tau -1}) +(u^{\tau -1}-u^{\tau -2}) \end{aligned}$$

for $\tau \ge 2$, and

$$\begin{aligned} u^{\tau ,0}-u^{\tau } =u^0-u^1 \end{aligned}$$

for $\tau =1$, by utilizing (7.4) we have the following inequalities:

$$\begin{aligned} \Vert u^{\tau ,0}-u^{\tau }\Vert _{\infty }\le & {} 2\Vert u^{\tau -1,k_o}-u^{\tau -1}\Vert _{\infty } +\Vert u^{\tau -2,k_o}-u^{\tau -2}\Vert _{\infty } \nonumber \\&+\,2 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \end{aligned}$$

(7.6)

if $\tau \ge 2$, and

$$\begin{aligned} \Vert u^{\tau ,0}-u^{\tau }\Vert _{\infty } \le \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \end{aligned}$$

(7.7)

if $\tau =1$.

The substitution of (7.6) into (7.5) directly results in the recurrence formula

$$\begin{aligned} \Vert u^{\tau ,k_o}-u^{\tau }\Vert _{\infty }\le & {} \left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} \right) \Vert u^{\tau -1,k_o}-u^{\tau -1}\Vert _{\infty } +\sigma ^{k_o} \Vert u^{\tau -2,k_o}-u^{\tau -2}\Vert _{\infty } \nonumber \\&+\,\sigma ^{k_o} \left[ 2 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \right] \nonumber \\\le & {} \left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} \right) \Vert u^{\tau -1,k_o}-u^{\tau -1}\Vert _{\infty } +\sigma ^{k_o} \Vert u^{\tau -2,k_o}-u^{\tau -2}\Vert _{\infty } \nonumber \\&+\,2 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \end{aligned}$$

(7.8)

for $\tau \ge 2$. As for $\tau =1$, it follows from (7.5) and (7.4) that

$$\begin{aligned} \Vert u^{1,k_o}-u^{1}\Vert _{\infty }\le & {} \sigma ^{k_o} \Vert u^{1,0}-u^{1}\Vert _{\infty } +\frac{1}{1+\eta _{\min } \, \triangle {t}} \Vert u^{0,k_o}-u^{0}\Vert _{\infty } \nonumber \\= & {} \sigma ^{k_o} \Vert u^{0}-u^{1}\Vert _{\infty } \nonumber \\\le & {} \sigma ^{k_o} \left[ \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \right] \nonumber \\= & {} \sigma ^{k_o} \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2), \end{aligned}$$

(7.9)

for $\tau =2$ the recurrence formula (7.8) particularly reads as

$$\begin{aligned} \Vert u^{2,k_o}-u^{2}\Vert _{\infty }\le & {} \left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} \right) \Vert u^{1,k_o}-u^{1}\Vert _{\infty } \nonumber \\&+\,2 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \nonumber \\\le & {} \left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} \right) \left[ \sigma ^{k_o} \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \right] \nonumber \\&+\,2 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \nonumber \\= & {} \left[ 2+\sigma ^{k_o} \left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} \right) \right] \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) \nonumber \\&+\,\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \nonumber \\\le & {} \left( 2+\sigma ^{k_o}\right) \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \nonumber \\\le & {} 3 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2). \end{aligned}$$

(7.10)

Here we have used the fact

$$\begin{aligned} 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} \le 1, \end{aligned}$$

which is readily implied by the restriction imposed on the positive integer $k_o$. We further remark that the restriction on $k_o$ is equivalent to the condition

$$\begin{aligned} 3\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}}< 1 \quad \text{ or }\quad \sigma ^{k_o} < \frac{\eta _{\min } \, \triangle {t}}{3 \left( 1+\eta _{\min } \, \triangle {t}\right) }, \end{aligned}$$

(7.11)

which is equivalent to $\theta <1$, too.

In addition, denote by

$$\begin{aligned} \tilde{\theta }=\frac{1}{2} \left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} -\sqrt{ \left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}}\right) ^2 +4\sigma ^{k_o} } \right) . \end{aligned}$$

Then, for $\tau \ge 3$, we can rewrite the second-order difference inequality (7.8) as

$$\begin{aligned} \Vert u^{\tau ,k_o}-u^{\tau }\Vert _{\infty } -\tilde{\theta } \Vert u^{\tau -1,k_o}-u^{\tau -1}\Vert _{\infty }\le & {} \theta \left( \Vert u^{\tau -1,k_o}-u^{\tau -1}\Vert _{\infty } -\tilde{\theta } \Vert u^{\tau -2,k_o}-u^{\tau -2}\Vert _{\infty } \right) \\&+\,2 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2), \end{aligned}$$

which, by induction, leads to

$$\begin{aligned} \Vert u^{\tau ,k_o}-u^{\tau }\Vert _{\infty }\le & {} \tilde{\theta } \Vert u^{\tau -1,k_o}-u^{\tau -1}\Vert _{\infty } +\theta ^{\tau -2} \left( \Vert u^{2,k_o}-u^{2}\Vert _{\infty } -\tilde{\theta } \Vert u^{1,k_o}-u^{1}\Vert _{\infty } \right) \nonumber \\&+\sum \limits _{i=0}^{\tau -3} \theta ^i \left[ 2 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \right] . \end{aligned}$$

(7.12)

Since $\tilde{\theta }<0$ and $\sqrt{1+s} \le 1+\frac{1}{2} s$ (for $\forall s \ge 0$), it holds that

$$\begin{aligned} |\tilde{\theta }|= & {} \frac{1}{2} \left[ \sqrt{ \left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} \right) ^2 +4\sigma ^{k_o} } -\left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} \right) \right] \\= & {} \frac{1}{2} \left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} \right) \left( \sqrt{1+ \frac{4\sigma ^{k_o}}{\left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}}\right) ^2} }-1 \right) \\\le & {} \frac{1}{2} \left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}} \right) \left[ \left( 1+ \frac{1}{2} \, \frac{4\sigma ^{k_o}}{\left( 2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}}\right) ^2} \right) -1 \right] \\= & {} \frac{\sigma ^{k_o}}{2\sigma ^{k_o}+\frac{1}{1+\eta _{\min } \, \triangle {t}}} \\\le & {} (1+\eta _{\min } \, \triangle {t}) \sigma ^{k_o}. \end{aligned}$$

Thereby, by making use of (7.9) and (7.10), from (7.12) we can obtain the estimate

$$\begin{aligned} \Vert u^{\tau ,k_o}-u^{\tau }\Vert _{\infty }\le & {} \theta ^{\tau -2} \left( \Vert u^{2,k_o}-u^{2}\Vert _{\infty } -\tilde{\theta } \Vert u^{1,k_o}-u^{1}\Vert _{\infty } \right) \nonumber \\&+\,\frac{1-\theta ^{\tau -2}}{1-\theta } \left[ 2 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \right] \nonumber \\\le & {} \theta ^{\tau -2} \left\{ 3 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \right. \nonumber \\&+\,(1+\eta _{\min } \, \triangle {t}) \sigma ^{k_o} \left. \left[ \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) \sigma ^{k_o} +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \right] \right\} \nonumber \\&+\,\frac{1-\theta ^{\tau -2}}{1-\theta } \left[ 2 \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \right] \nonumber \\= & {} \left[ \frac{2-3\theta ^{\tau -1}+\theta ^{\tau -2}}{1-\theta } +\left( 1+\eta _{\min } \, \triangle {t}\right) \sigma ^{2k_o} \right] \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) \nonumber \\&+\,\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \nonumber \\\le & {} \left( \frac{3}{1-\theta } +\eta _{\min }^2 \, \triangle {t^2} \right) \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \nonumber \\= & {} \frac{3}{1-\theta } \left( \gamma _0 \, \triangle {t} +\frac{2\varepsilon }{\eta _{\min }} \right) +\mathscr {O}(\triangle {t^2})+\mathscr {O}(h^2) \end{aligned}$$

(7.13)

for $\tau \ge 3$. Here the last inequality has been derived by employing the bounds

$$\begin{aligned} 2-3\theta ^{\tau -1}+\theta ^{\tau -2} \le 3 \left( 1-\theta ^{\tau -1}\right) \le 3 \end{aligned}$$

and

$$\begin{aligned} \left( 1+\eta _{\min } \, \triangle {t}\right) \sigma ^{2k_o}< & {} \left( 1+\eta _{\min } \, \triangle {t}\right) \left[ \frac{\eta _{\min } \, \triangle {t}}{3\left( 1+\eta _{\min } \, \triangle {t}\right) } \right] ^2 \\= & {} \frac{\eta _{\min }^2 \, \triangle {t^2}}{9\left( 1+\eta _{\min } \, \triangle {t}\right) } \\< & {} \eta _{\min }^2 \, \triangle {t^2}; \end{aligned}$$

see (7.11).

Recalling (7.9) and (7.10), we see that (7.13) holds true for all $\tau \ge 1$, too. Now, the error bound (4.3) follows straightforwardly by substitutions of both (7.3) and (7.13) into (7.1). $\Box$