Compact Crank–Nicolson Schemes for a Class of Fractional Cattaneo Equation in Inhomogeneous Medium

Abstract

In this paper, a compact Crank–Nicolson scheme is proposed and analyzed for a class of fractional Cattaneo equation. In developing the scheme, the Crank–Nicolson discretization is applied for the time derivatives both in classical and in fractional definitions. Moreover, a compact operator for the spatial derivative involving variable coefficient is derived. When the variable coefficient is replaced by a unit constant, it reveals a particularly significant situation that the derived compact operator degenerates to the common four-order compact operator for Laplacian. It is proved that the scheme is stable and convergent in $H^1$ semi-norm via energy method. The convergence orders are $3-\gamma$ in time and 4 in space, where $\gamma \in (1,2)$ is the order of fractional derivative. In addition, a compact Crank–Nicolson alternating direction implicit (ADI) scheme is constructed for the 2D case and the corresponding theoretical analysis is also presented. The derived ADI scheme combines the discretization operators for time both in classical and in fractional forms, which allows the utilization of the modified ADI scheme to reduce the storage requirements and the consumption of CPU time. The applicability and the accuracy of the scheme are demonstrated by numerical experiments in 1D and 2D cases.

Appendix

Before we prove Lemma 2.3, we give several formulae.

Lemma 6.1

Let $g(x)\in \mathcal {C}^{2l+1}[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}],$ then it holds that

$$\begin{aligned} \frac{g(x_{i+\frac{1}{2}})-g(x_{i-\frac{1}{2}})}{h}= \sum _{n=0}^{l-1}\frac{1}{(2n+1)!}\left( \frac{h}{2}\right) ^{2n}g^{(2n+1)}(x_i)+R^l[g]_{i,\frac{h}{2}}, \end{aligned}$$

where

$$\begin{aligned} R^l[g]_{i,\frac{h}{2}}=\frac{1}{2(2l)!}\left( \frac{h}{2}\right) ^{2l}\int \limits _{0}^{1} \left[ g^{(2l+1)}(x_{i}+\frac{h}{2}s)+g^{(2l+1)}(x_{i}-\frac{h}{2}s)\right] (1-s)^{2l}ds, \end{aligned}$$

and there exists a positive constant $C_l$, which depends on the regularity of function $g(x)$, such that

$$\begin{aligned} \left| R^l[g]_{i,\frac{h}{2}}\right| \leqslant C_l h^{2l}. \end{aligned}$$

The notation $R^l[\Theta ]_{\Phi (i),\Psi (h)}$ denotes the remainder term of the asymptotic expansion of the central difference quotient of function $\Theta$ at the point $x_{\Phi (i)}$ with step size $\Psi (h).$

The approximations in the above lemma with cases $l=1,\,2$ will be utilized in the following proof. We denote $C_1,\,C_2$ two generic constants which are independent of space step $h$ but rely on the regularity of the function whose derivative is approximated, and the value of which may vary at each occurrence. In addition, the following constant $C$ is defined and used in the similar way.

Lemma 6.2

[34, 45] Let $g(x)\in \mathcal {C}^6[x_{i-1},x_{i+1}],$ then it holds that

$$\begin{aligned} \frac{g(x_{i+1})-2g(x_{i})+g(x_{i-1})}{h^2}= g''(x_{i})+R[g]_{i,h}, \end{aligned}$$

where

$$\begin{aligned} R[g]_{i,h}=\frac{h^2}{6}\int \limits _{0}^{1} \left[ g^{(4)}(x_{i}+hs)+g^{(4)}(x_{i}-hs)\right] (1-s)^3ds, \end{aligned}$$

then there exists a positive constant $C$, which depends on the regularity of function $g(x)$, such that

$$\begin{aligned} \left| R[g]_{i,h}\right| \leqslant C h^{2}. \end{aligned}$$

The above two lemmas are derived by using Taylor expansion with integral remainders. By utilizing the above Lemmas, we complete the proof of Lemma 2.3 in the following.

Proof of Lemma 2.3

We reclaim that the variable coefficient function $D(x)$ satisfies the smoothness assumptions made thought out the paper. Starting from using Lemma 6.1 with $l=2,$ we derive the main discrete equation:

$$\begin{aligned} F_i&=\frac{1}{h}\left[ D(x_{i+\frac{1}{2}})g_{x}(x_{i+\frac{1}{2}})-D(x_{i-\frac{1}{2}})g_{x}(x_{i-\frac{1}{2}})\right] -\frac{h^2}{24}f_{xx}(x_i)+R^2[Dg_{x}]_{i,\frac{h}{2}}\nonumber \\&=\frac{1}{h}\left[ D(x_{i+\frac{1}{2}})\left( \delta _{x} G_{i+\frac{1}{2}}-\frac{h^2}{24}g_{xxx}(x_{i+\frac{1}{2}})+R^2[g]_{i+\frac{1}{2},\frac{h}{2}}\right) \right. \nonumber \\&\quad \left. -D(x_{i-\frac{1}{2}})\left( \delta _{x} G_{i-\frac{1}{2}}-\frac{h^2}{24}g_{xxx}(x_{i-\frac{1}{2}})+R^2[g]_{i-\frac{1}{2},\frac{h}{2}}\right) \right] -\frac{h^2}{24}f_{xx}(x_i)+R^2[Dg_{x}]_{i,\frac{h}{2}}, \end{aligned}$$

(6.1)

where

$$\begin{aligned} \left| R^2[\Theta ]_{\Phi (i),\Psi (h)}\right| \leqslant C_2 h^4. \end{aligned}$$

(6.2)

$\square$

Next, (6.1) is rewritten as

$$\begin{aligned} F_i&=\delta _{x}(D\delta _{x}G)_i-\frac{h^2}{24}\cdot \frac{1}{h}\left[ D(x_{i+\frac{1}{2}})g_{xxx}(x_{i+\frac{1}{2}})-D(x_{i-\frac{1}{2}})g_{xxx}(x_{i-\frac{1}{2}})\right] \nonumber \\&\quad +\!\frac{1}{h}\left[ D(x_{i\!+\!\frac{1}{2}})R^2[g]_{i\!+\!\frac{1}{2},\frac{h}{2}}\!-\!D(x_{i-\frac{1}{2}})R^2[g]_{i-\frac{1}{2},\frac{h}{2}}\right] \!-\!\frac{h^2}{24}(\delta _{x}^2 F_i\!+\!R[f]_{i,h}) \!+\!R^2[Dg_{x}]_{i,\frac{h}{2}}, \end{aligned}$$

(6.3)

where Lemma 6.2 is used to approximate the second order derivative of $f$ in (6.1) and the bound for the truncation error $R[f]_{i,h}$ is

$$\begin{aligned} \left| R[f]_{i,h}\right| \leqslant C h^2. \end{aligned}$$

(6.4)

As similar calculations conducted in [35], we gain the following equality

$$\begin{aligned} D(x)g_{xxx}(x)=f_{x}(x)-2\frac{D_{x}(x)}{D(x)}f(x)+2D_1(x)g_{x}(x). \end{aligned}$$

Owing to the above equality, we can treat the second term at the right hand side of (6.3) as

$$\begin{aligned}&\frac{1}{h}\left[ D(x_{i+\frac{1}{2}})g_{xxx}(x_{i+\frac{1}{2}})-D(x_{i-\frac{1}{2}})g_{xxx}(x_{i-\frac{1}{2}})\right] \nonumber \\&\quad =\!\frac{1}{h}\left[ \left( f_{x}(x)\!-\!2\frac{D_{x}(x)}{D(x)}f(x)\!+\!2{D_1}g_{x}(x)\right) \big |_{x_{i\!+\!\frac{1}{2}}} \!-\!\left( f_{x}(x)\!-\!2\frac{D_{x}(x)}{D(x)}f(x)\!+\!2{D_1}g_{x}(x)\right) \big |_{x_{i\!-\!\frac{1}{2}}}\right] \nonumber \\&\quad =\delta _{x}^2F_i-2\Delta _{x}\left( \frac{D_{x}}{D}F\right) _i+2\delta _{x}({D_1}\delta _{x}G)_i+\frac{1}{h}\left( R^1[f]_{i+\frac{1}{2},h}-R^1[f]_{i-\frac{1}{2},h}\right) \nonumber \\&\qquad -\!2\left( R^1\left[ \frac{D_{x}}{D}f\right] _{i,h}\!-\!R^1\left[ \frac{D_{x}}{D}f\right] _{i,\frac{h}{2}}\right) \!+\!\frac{2}{h}\left( D_1(x_{i+\frac{1}{2}})R^1[g]_{i+\frac{1}{2},h}\!-\!D_1(x_{i-\frac{1}{2}})R^1[g]_{i-\frac{1}{2},h}\right) .\nonumber \\ \end{aligned}$$

(6.5)

From Lemma 6.1, $R^1[\Theta ]_{\Phi (i),\Psi (h)}$ in the above equality is bounded by

$$\begin{aligned} \left| R^1[\Theta ]_{\Phi (i),\Psi (h)}\right| \leqslant C_1 h^2. \end{aligned}$$

(6.6)

Substituting (6.5) into (6.3) we obtain

$$\begin{aligned} F_i=\delta _{x}(D\delta _{x}G)_i-\frac{h^2}{24}\left[ \delta _{x}^2F_i-2\Delta _{x}\left( \frac{D_{x}}{D}F\right) _i+2\delta _{x}({D_1}\delta _{x}G)_i\right] -\frac{h^2}{24}\delta _{x}^2 F_i+r_i, \end{aligned}$$

(6.7)

i.e.

$$\begin{aligned} F_{i}+\frac{h^2}{12}\left[ \delta _{x}^2F_{i}-\Delta _{x}\left( \frac{D_1}{D}F\right) _{i}\right] =\delta _{x}(D_2\delta _{x}G)_i+r_i, \end{aligned}$$

(6.8)

where

$$\begin{aligned} r_i&=-\frac{h^2}{24}\left[ \frac{1}{h}\left( R^1[f]_{i+\frac{1}{2},h}-R^1[f]_{i-\frac{1}{2},h}\right) -2\left( R^1\left[ \frac{D_{x}}{D}f\right] _{i,h}-R^1\left[ \frac{D_{x}}{D}f\right] _{i,\frac{h}{2}}\right) \right. \nonumber \\&\quad \left. +\frac{2}{h}\left( D_1(x_{i+\frac{1}{2}})R^1[g]_{i+\frac{1}{2},h}-D_1(x_{i-\frac{1}{2}})R^1[g]_{i-\frac{1}{2},h}\right) \right] \nonumber \\&\quad +\frac{1}{h}\left[ D(x_{i+\frac{1}{2}})R^2[g]_{i+\frac{1}{2},\frac{h}{2}}-D(x_{i-\frac{1}{2}})R^2[g]_{i-\frac{1}{2},\frac{h}{2}}\right] -\frac{h^2}{24}R[f]_{i,h}+R^2[Dg_{x}]_{i,\frac{h}{2}}\nonumber \\&=\!\!-\!\frac{h^2}{24}\left[ \text {(I)}\!-\!2\left( R^1\left[ \frac{D_{x}}{D}f\right] _{i,h}\!-\!R^1\left[ \frac{D_{x}}{D}f\right] _{i,\frac{h}{2}}\right) \!\!+\!\!\text {(II)}\right] \!+\!\text {(III)}\!-\!\frac{h^2}{24}R[f]_{i,h}\!+\!R^2[bg_{x}]_{i,\frac{h}{2}}. \end{aligned}$$

(6.9)

Here we analyze $\text {(I)}$ in the following

$$\begin{aligned} \text {(I)}&=\frac{1}{h}\left( R^1[f]_{i+\frac{1}{2},h}-R^1[f]_{i-\frac{1}{2},h}\right) \nonumber \\&=\frac{h}{2}\int \limits _{0}^{1} \left[ f^{(3)}(x_{i+\frac{1}{2}}+hs)+f^{(3)}(x_{i+\frac{1}{2}}-hs)\right] (1-s)^2ds\nonumber \\&\quad -\frac{h}{2}\int \limits _{0}^{1} \left[ f^{(3)}(x_{i-\frac{1}{2}}+hs)+f^{(3)}(x_{i-\frac{1}{2}}-hs)\right] (1-s)^2ds\nonumber \\&=\frac{h}{2}\int \limits _{0}^{1} \left[ f^{(3)}(x_{i+\frac{1}{2}}+hs)-f^{(3)}(x_{i-\frac{1}{2}}+hs)\right] (1-s)^2ds\nonumber \\&\quad +\frac{h}{2}\int \limits _{0}^{1} \left[ f^{(3)}(x_{i+\frac{1}{2}}-hs)-f^{(3)}(x_{i-\frac{1}{2}}-hs)\right] (1-s)^2ds\nonumber \\&=\frac{h^2}{2}\left[ \int \limits _{0}^{1}\int \limits _{0}^{1}f^{(4)}(x_{i-\frac{1}{2}}+(\lambda _1+s)h)(1-s)^2d\lambda _1 ds\right. \nonumber \\&\qquad \quad \quad \ \ +\left. \int \limits _{0}^{1}\int \limits _{0}^{1}f^{(4)}(x_{i-\frac{1}{2}}+(\lambda _2-s)h)\lambda _2 ds\right] , \end{aligned}$$

(6.10)

then we obtain

$$\begin{aligned} |\text {(I)}|\leqslant C_1'h^2, \end{aligned}$$

where $C_1'$ is independent of the step size $h$.

Resort to the same methodology in analysis of $\text {(I)}$, the bounds of $\text {(II)}$ and $\text {(III)}$ are reached in the following

$$\begin{aligned} |\text {(II)}|&\leqslant C_2'h^2,\\ |\text {(III)}|&\leqslant C_3'h^4. \end{aligned}$$

Noticing (6.2), (6.4) and (6.6), the bounds of last three terms in (6.9) are obviously shown. We finish the proof of Lemma 2.3 by substituting the bounds of $\text {(I)}$, $\text {(II)}$ and $\text {(III)}$ into (6.9).