A spline collocation method for a fractional mobile–immobile equation with variable coefficients

Abstract

The Crank–Nicolson orthogonal spline collocation (OSC) methods are considered for approximate solution of the variable coefficient fractional mobile–immobile equation. The convection, diffusion, and reaction coefficients can depend on both the spatial and temporal variables, simultaneously. Combining with Crank–Nicolson scheme and weighted and shifted Grünwald difference approximation in time, we establish OSC method in space. It is proved that our proposed fully methods are of optimal order in certain $H_j$ ($j=0,1$) norms. Moreover, we derive $L^{\infty }$ estimates in space. Numerical results are also provided to verify our proposed algorithm.

1 Introduction

In this paper, we consider the following fractional-order mobile–immobile equation with variable coefficients:

$$\begin{aligned} ^C_0D^{\alpha }_tu+u_t+{\mathcal {L}}u= & {} f(x,y,t), (x,y,t)\in \Omega _T\equiv \Omega \times (0,T], \end{aligned}$$

(1.1)

$$\begin{aligned} u(x,y,0)= & {} u_0(x,y),\quad (x,y)\in \Omega ,\end{aligned}$$

(1.2)

$$\begin{aligned} u(x,y,t)= & {} 0,\quad (x,y,t)\in \partial \Omega \times (0,T]. \end{aligned}$$

(1.3)

where $\Omega \subset \text {R}^2$ is bounded convex polygonal domain with boundary $\partial \Omega$, and f and $u_0$ are given functions. ${\mathcal {L}}u={\mathcal {L}}_1u+{\mathcal {L}}_2u$, ${\mathcal {L}}_1u=-p_1(x,y,t)u_{xx}+q_1(x,y,t)u_x+r(x,y,t)u$, and ${\mathcal {L}}_2u=-p_2(x,y,t)u_{yy}+q_2(x,y,t)u_y$. There exist positive constants $p_{\text {min}}, p_{\text {max}}$, such that $0< p_{\text {min}} \le p_1(x,y,t),p_2(x,y,t)\le p_{\text {max}}$. Herein, we consider operator ${\mathcal {L}}$ in the non-divergence forms rather than in the divergence forms, because the non-divergence forms are more natural for OSC spatial discretization. The Caputo fractional derivative $^C_0D^{\alpha }_t$ is defined by:

$$\begin{aligned} ^C_0D^{\alpha }_tu(\cdot ,t) =\frac{1}{\Gamma (1-\alpha )}\int _0^t\frac{\partial u(\cdot ,s)}{\partial s}\frac{{\text {d}}s}{(t-s)^{\alpha }},\quad 0<\alpha <1. \end{aligned}$$

(1.4)

The fractional-order mobile–immobile equations are a type of second order PDEs, which describe a family of problems including heat diffusion and ocean acoustic propagation in mathematical systems with the time variable t and behaves like heat diffusing through a solid. The time drift term $u_t$ is added to exhibit the motion time and thus helps to distinguish the status of particles conveniently. The model is the limiting equation which control continuous time random walks with heavy-tailed random waiting times. Hence, it is difficult or infeasible to find the analytical solution of this equations in most cases, and then to find its numerical solutions become more necessary. Most of previous works concentrate on constant coefficient problems Jiang (2015), Wei (2017, 2018), Chen et al. (2016), He and Pan (2017, 2018), and Liu et al. (2015). For variable coefficient, Cui (2015) studied the time fractional convection–diffusion reaction equation with variable coefficients by the compact exponential scheme. Wang et al. (2019) provide a novel high-order approximate scheme for time-fractional 2D diffusion equations with variable coefficient. Liu et al. (2012) analyzed novel and efficient numerical methods for a class of fractional advection–dispersion models, including the mobile/immobile time-fractional advection–dispersion model with a Caputo fractional derivative. Subsequently, Liu et al. (2014) constructed an RBF meshless method for a fractal mobile–immobile transport model. Zhang et al. (2013) described an implicit Euler approximation for the time-variable fractional-order mobile–immobile advection–dispersion model. Recently, Liu et al. Liu and Li (2018) introduced the Crank–Nicolson finite-difference scheme to solve a time-variable fractional-order mobile–immobile advection–dispersion equation, and proved a priori estimates of discrete $L^2$-norm.

Published articles on numerical methods for fractional mobile–immobile convection–subdiffusion equation with variable coefficients are still sparse. This motivates us to consider high accuracy numerical schemes for solving them. The current work is devoted to deriving a high-order scheme by combining Crank–Nicolson and weighted and shifted Grünwald difference approximation for time derivative and OSC scheme for space. There have been many earlier research papers discussing OSC schemes for steady and/or unsteady convection–diffusion equations of integer order, e.g., Bialecki (1998), Bialecki and Fernandes (1993), Fernandes and Fairweather (1993), Yan and Fairweather (1992), Zhang et al. (2019), and Yang et al. (2019). However, numerical approximation referring OSC method for fractional-order convection–subdiffusion equations with variable coefficients is still at an early stage of development. Thus, it is important and necessary to develop efficient numerical methods to solve them.

The structure of the paper is organized as follows. In Sect. 2, the Crank–Nicolson OSC method is derived. The heart of our paper is Sect. 3, where we prove the stability and convergence in certain $H_j$ ($j=0,1$) norms for proposed scheme. In Sect. 4, numerical experiments are given; at last, some conclusions are drawn in Sect. 5.

2 The Crank–Nicolson OSC scheme

2.1 Preliminaries

Let $N_x, N_y$, and N be some positive integer, the collection of spatial quasi-uniform Percell and Wheeler (1980) mesh of $\Omega$ defined by $\delta \equiv \delta _x\times \delta _y, \delta _x: 0=x_0<x_1<\cdots<x_{N_x}=1, \delta _y: 0=y_0<y_1<\cdots <y_{N_y}=1$, $1\le k\le N_x, 1\le l\le N_y$.

Denote by ${\mathcal {M}}_r(\delta )\equiv {\mathcal {M}}(r, \delta _x)\otimes {\mathcal {M}}(r, \delta _y)$ a space of piecewise polynomials in x and y, ${\mathcal {M}}(r,\delta _x)=\left\{ u|u\in C^1([0,1]), u|_{[x_{k-1},x_{k}]}\in P_r, u(0)=u(1)=0\right\}$, and $P_r$ denotes the space of all polynomials of degree less than or equal to r. With ${\mathcal {M}}(r,\delta _y)$ defined similarly.

Let $\{\lambda _k\}_{k=1}^{r-1}$ and $\{\omega _k\}_{k=1}^{r-1}$ be the nodes and weights of the $(r-1)$-point Gauss quadrature rule on [0, 1]. In domain $\Omega$, we define Gauss collocation points set: $\Lambda _r\equiv \{\xi |\xi =(\xi ^x, \xi ^y), \xi ^x\in \Lambda _x, \xi ^y\in \Lambda _y\},\Lambda _x=\{x_{i-1}+\lambda _kh^x_i\}_{i,k=1}^{N_x,r-1}$, $h_k^x=x_{k}-x_{k-1}$. With $\Lambda _y$ defined similarly.

At last, the discrete inner product and norm are defined by:

$$\begin{aligned}&\left\langle U,V\right\rangle =\sum \limits _{i=1}^{N_x}\sum \limits _{j=1}^{N_y} h_i^xh_j^y\sum \limits _{k=1}^{r-1}\sum \limits _{l=1}^{r-1}\omega _k \omega _l(UV)(\xi _{i,k}^x,\xi _{j,l}^y),\quad U,V\in {\mathcal {M}}_r(\delta ),\\&\Vert V\Vert _{{\mathcal {M}}_r}^2=\langle V, V\rangle , \qquad V\in {\mathcal {M}}_r(\delta ). \end{aligned}$$

2.2 Construction of OSC scheme

In this subsection, we will consider Crank–Nicolson OSC scheme for approximating the solution of problem (1.1). Let temporal domain [0, T] be divided by the partition $\{t_k\}_{k=0}^K$ with $t_k=k\tau$, and $\tau =T/K$. Next, we introduce some difference quotient notations:

$$\begin{aligned} V^n(\cdot ,\cdot )=V(\cdot ,\cdot ,t_n), \delta _t V^{n+1}=\frac{V^{n+1}-V^{n}}{\tau },\quad V^{n+\frac{1}{2}}=\frac{1}{2}(V^{n+1}+V^{n}). \end{aligned}$$

We first consider the weighted and shifted Grünwald–Letnikon approximation Tian et al. (2015) and Wang and Vong (2014) for $^C_0D^{\alpha }_tu(\cdot ,t)$:

$$\begin{aligned}&^C_0D^{\alpha }_tu(x,y,t_{n+1}) =\tau ^{-\alpha }\sum \limits _{k=0}^{n+1}\lambda _k^{(\alpha )} u(x,y,t_{n+1-k})+R_{(\alpha )}^{n+1}, \end{aligned}$$

(2.1)

where

$$\begin{aligned} \left\{ \begin{array}{ll} \lambda _k^{(\alpha )}=-\frac{\alpha }{2}g_{k-1}^{(\alpha )}+\frac{2+\alpha }{2}g_k^{(\alpha )}, &{}\quad {k = 1,2,3\ldots ;} \\ \lambda _0^{(\alpha )}=\frac{2+\alpha }{2}g_0^{(\alpha )} , &{}\quad {k=0.} \end{array} \right. \end{aligned}$$

(2.2)

and

$$\begin{aligned} g_k^{(\alpha )}=\left( 1-\frac{\alpha +1}{k}\right) g_{k-1}^{(\alpha )}, \quad g_0^{(\alpha )}=1. \end{aligned}$$

It can be checked directly for $0< \alpha < 1$ that the coefficients $\{g_k^{(\alpha )}\}_{k=0}^{\infty }$ and $\{\lambda _k^{(\alpha )}\}_{k=0}^{\infty }$ satisfy the following properties:

$$\begin{aligned} \left\{ \begin{array}{ll} g_1^{(\alpha )}=-\alpha <0, &{} g_2^{(\alpha )}\le g_3^{(\alpha )}\le g_4^{(\alpha )}\le \cdots \le 0;\\ \sum \limits _{k=1}^{\infty }g_k^{(\alpha )}=-1, &{}\sum \limits _{k=0}^{n}g_k^{(\alpha )}\ge 0,\quad n\ge 1; \\ \lambda _0^{(\alpha )}=1+\frac{\alpha }{2}>0, &{}\quad \sum \limits _{k=0}^{n+1}|\lambda _k^{(\alpha )}|\le 2\alpha +2. \end{array} \right. \end{aligned}$$

(2.3)

Moreover, for any real vector $(w_1, w_2, \ldots , w_k)^{\text {T}}\in {\mathbb {R}}^k$, it holds that:

$$\begin{aligned} \sum \limits _{n=0}^{k-1}\left( \sum \limits _{p=0}^{n}\lambda _p^{(\alpha )}w_{n+1-p}\right) w_{n+1}\ge 0,\quad \ k=1,2,\ldots . \end{aligned}$$

(2.4)

For the proof, see Wang and Vong (2014).

The estimate of $R^{(\alpha )}_{n+1}$ can be found in Tian et al. (2015), and satisfies:

$$\begin{aligned} \left| R_{(\alpha )}^{n+1}\right| \le C\tau ^2\left\| {\mathfrak {F}}[^{RL}_0D^{\alpha +2}_tu ](\omega )\right\| _{L^1}, \end{aligned}$$

(2.5)

where ${\mathfrak {F}}$ denotes the Fourier transform symbol, and $u\in C^2$, $^{RL}_0D^{\alpha +2}_tu$, and its Fourier transform belong to $L^{1}({\mathbb {R}})$.

Therefore, using the approximate formula (2.1), the Crank–Nicolson OSC scheme for Eq. (1.1) consists in finding $\{u_h^n\}_{n=0}^{K}\subset {\mathcal {M}}_r(\delta )$, such that, for all $\xi \in \Lambda _r$:

$$\begin{aligned} \left\{ \delta _tu_h^{n+1}+\frac{\tau ^{-\alpha }}{2}\left[ \sum \limits _{k=0}^{n+1}\lambda _k^{(\alpha )}u_h^{n+1-k} +\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}u_h^{n-k}\right] +{\mathcal {L}}^{n+\frac{1}{2}}u_h^{n+\frac{1}{2}}\right\} (\xi ) =f^{n+\frac{1}{2}}(\xi ), \end{aligned}$$

where ${\mathcal {L}}^{n+\frac{1}{2}}$ and $f^{n+\frac{1}{2}}$ denote the operator ${\mathcal {L}}(t)$ and the function f(t), respectively, evaluated at $t=t_{n+\frac{1}{2}}$. For the stability and error analysis, we rewrite the above equation in the equivalent form:

$$\begin{aligned}&\langle \delta _tu_h^{n+1},v_h\rangle +\langle {\mathcal {L}}^{n+\frac{1}{2}}u_h^{n+\frac{1}{2}},v_h\rangle =-\tau ^{-\alpha }\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\langle u_h^{n-k+\frac{1}{2}},v_h\rangle \nonumber \\&\quad -\frac{\tau ^{-\alpha }}{2}\lambda _{n+1}^{(\alpha )}\langle u_h^{0},v_h\rangle +\langle f^{n+\frac{1}{2}},v_h\rangle , \quad v_h\in {\mathcal {M}}_r(\delta ),\ 0\le n\le K-1, \end{aligned}$$

(2.6)

where, for convenience, we have omitted the dependence of $u^{n+1}(\xi )$ on $(\xi )$ in the above equation.

3 Analysis of the Crank–Nicolson OSC scheme

To analyze the convergence of fully discrete scheme (2.6), we begin with the following Lemma.

Lemma 3.1

Bialecki and Fernandes (1993) If ${\mathcal {L}}={\mathcal {L}}_1+{\mathcal {L}}_2$, and assume $p_1\in C^{5,0,0}(\Omega _T)$, $p_2\in C^{0,5,0}(\Omega _T)$, $q_1,q_2,r\in C(\Omega _T)$. Also assume that $p_i,i=1,2$ satisfy the Lipschitz condition with respect to t, that is, for $(x,y)\in \Omega , t_1,t_2\in (0,T]$, there is a constant $C>0$, such that

$$\begin{aligned} |p_i(x,y,t_1)-p_i(x,y,t_2)|\le C|t_1-t_2|,\quad i=0,1, \end{aligned}$$

then we can show that:

$$\begin{aligned} \langle {\mathcal {L}}(t)W,V\rangle =A_0(t;W,V)+A_1(t;W,V), t\in (0,T], W, V\in {\mathcal {M}}_r(\delta ), \end{aligned}$$

(3.1)

where $A_i(t;\cdot ,\cdot )$, $t\in (0,T]$, $i=0,1$, are real-valued bilinear forms on ${\mathcal {M}}_r(\delta )\times {\mathcal {M}}_r(\delta )$ for all $t\in (0,T]$, $W,V\in {\mathcal {M}}_r(\delta )$, $p_{min}$, $p_{max}$, and C are positive constants, we have:

$$\begin{aligned} (1)\ A_0(t;W,V)= & {} A_0(t;V,W); \end{aligned}$$

(3.2)

$$\begin{aligned} (2)\ p_{min}\langle -\Delta W,W\rangle \le A_0(t;W,W)\le & {} p_{max}\langle -\Delta W,W\rangle ; \end{aligned}$$

(3.3)

$$\begin{aligned} (3)\ |A_0(t_1;W,W)-A_0(t_2;W,W)|\le & {} C|t_1-t_2|\langle -\Delta W,W\rangle ; \end{aligned}$$

(3.4)

$$\begin{aligned} (4)\ A_1(t_1;W,V)\le & {} C_{\varrho }\langle -\Delta W,W\rangle ^{\frac{1}{2}}\Vert V\Vert _{{\mathcal {M}}_r}; \end{aligned}$$

(3.5)

where

$$\begin{aligned} \varrho =\Vert q_1\Vert _{C(\Omega _T)}+\Vert q_2\Vert _{C(\Omega _T)}+\Vert r\Vert _{C(\Omega _T)}+\max \limits _{1\le i \le 5}\left( \Vert \frac{\partial ^{i}p_1 }{\partial x^{i}}\Vert _{C(\Omega _T)}, \Vert \frac{\partial ^{i}p_2 }{\partial y^{i}}\Vert _{C(\Omega _T)}\right) . \end{aligned}$$

For the proof, see Bialecki and Fernandes (1993), Lemma 3.2.

3.1 $L^2$ stability analysis

The $L^2$ stability of Crank–Nicolson OSC scheme (2.6) is given in the following theorem.

Theorem 3.2

The Crank–Nicolson OSC scheme (2.6) is stable with respect to $L^2$ norm. Specifically, for $u^m_h\in {\mathcal {M}}_r(\delta )$, it holds:

$$\begin{aligned} \left\| u_h^{m}\right\| ^2_{{\mathcal {M}}_r} \le C\left( \tau ^{1-\alpha }\left\| u_h^{0}\right\| ^2_{{\mathcal {M}}_r} +\tau \sum \limits _{n=0}^{m-1}\Vert f^{n+\frac{1}{2}}\Vert ^2_{{\mathcal {M}}_r}\right) , \quad 1\le m\le K. \end{aligned}$$

(3.6)

Proof

Taking $v_h=u_h^{n+\frac{1}{2}}$ in (2.6), for $0\le n\le K-1$, we obtain:

$$\begin{aligned}&\langle \delta _t u_h^{n+1},u_h^{n+\frac{1}{2}}\rangle +\langle {\mathcal {L}}^{n+\frac{1}{2}}u_h^{n+\frac{1}{2}},u_h^{n+\frac{1}{2}}\rangle \nonumber \\&\quad =-\tau ^{-\alpha }\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\langle u_h^{n-k+\frac{1}{2}},u_h^{n+\frac{1}{2}}\rangle \nonumber \\&\qquad -\frac{\tau ^{-\alpha }}{2}\lambda _{n+1}^{(\alpha )}\langle u_h^{0},u_h^{n+\frac{1}{2}}\rangle +\langle f^{n+\frac{1}{2}},u_h^{n+\frac{1}{2}}\rangle . \end{aligned}$$

(3.7)

Since

$$\begin{aligned} \left\langle \delta _t u_h^{n+1},u_h^{n+\frac{1}{2}}\right\rangle = \frac{1}{2}\delta _t \left\| u_h^{n+1}\right\| ^2_{{\mathcal {M}}_r}, \end{aligned}$$

(3.8)

it follows from (3.1) of Lemma 3.1 that:

$$\begin{aligned} \left\langle {\mathcal {L}}^{n+\frac{1}{2}}u_h^{n+\frac{1}{2}},u_h^{n+\frac{1}{2}}\right\rangle =A_0\left( t_{n+\frac{1}{2}};u_h^{n+\frac{1}{2}},u_h^{n+\frac{1}{2}}\right) +A_1\left( t_{n+\frac{1}{2}};u_h^{n+\frac{1}{2}},u_h^{n+\frac{1}{2}}\right) , \end{aligned}$$

(3.9)

from Eq. (3.4) of Fernandes and Fairweather (1993), we have:

$$\begin{aligned} \left\langle -\Delta u_h^{n+\frac{1}{2}},u_h^{n+\frac{1}{2}}\right\rangle \; \ge \; C\left\| \nabla u_h^{n+\frac{1}{2}}\right\| ^2\; \ge \; 0. \end{aligned}$$

(3.10)

Furthermore, using (3.3) and (3.5), we have:

$$\begin{aligned}&\left\langle {\mathcal {L}}^{n+\frac{1}{2}}u_h^{n+\frac{1}{2}},u_h^{n+\frac{1}{2}}\right\rangle \nonumber \\&\quad \ge p_{\text {min}}\left\langle -\Delta u_h^{n+\frac{1}{2}},u_h^{n+\frac{1}{2}}\right\rangle -C_{\varrho }\left\langle -\Delta u_h^{n+\frac{1}{2}},u_h^{n+\frac{1}{2}}\right\rangle ^{\frac{1}{2}}\Vert u_h^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r} \nonumber \\&\quad \ge p_{\text {min}}\Vert \nabla u_h^{n+\frac{1}{2}}\Vert ^2 -C_{\varrho }\Vert \nabla u_h^{n+\frac{1}{2}}\Vert \ \Vert u_h^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r} \nonumber \\&\quad \ge \frac{p_{\text {min}}}{2}\Vert \nabla u_h^{n+\frac{1}{2}}\Vert ^2 -C\Vert u_h^{n+\frac{1}{2}}\Vert ^2_{{\mathcal {M}}_r}; \end{aligned}$$

(3.11)

on substituting (3.8) and (3.11) into (3.7), multiplying the result equation by $2\tau$, and then summing from $n=0$ to $n=m-1$, $1\le m\le K$, we obtain:

$$\begin{aligned}&\left\| u_h^{m}\right\| ^2_{{\mathcal {M}}_r} +\tau p_{\text {min}} \sum \limits _{n=0}^{m-1}\Vert \nabla u_h^{n+\frac{1}{2}}\Vert ^2 \nonumber \\&\quad \le \left\| u_h^{0}\right\| ^2_{{\mathcal {M}}_r}+C\tau \sum \limits _{n=0}^{m-1}\Vert u_h^{n+\frac{1}{2}}\Vert ^2 -2\tau ^{1-\alpha }\sum \limits _{n=0}^{m-1}\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\left\langle u_h^{n-k+\frac{1}{2}},u_h^{n+\frac{1}{2}}\right\rangle \nonumber \\&\qquad -\tau ^{1-\alpha }\sum \limits _{n=0}^{m-1}\lambda _{n+1}^{(\alpha )}\left\langle u_h^{0},u_h^{n+\frac{1}{2}}\right\rangle +2\tau \sum \limits _{n=0}^{m-1}\left\langle f^{n+\frac{1}{2}},u_h^{n+\frac{1}{2}}\right\rangle . \end{aligned}$$

(3.12)

It follows from (2.4), we obtain:

$$\begin{aligned} -2\tau ^{1-\alpha }\sum \limits _{n=0}^{m-1}\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\left\langle u_h^{n-k+\frac{1}{2}},u_h^{n+\frac{1}{2}}\right\rangle \le 0, \end{aligned}$$

dropping the non-positive the third term on the RHS of (3.12), then applying the Cauchy–Schwarz inequality and Young’s inequality to the last two terms on the RHS of the resulting expression, and noticing $\Vert u_h^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r}\le \frac{1}{2}(\Vert u_h^{n+1}\Vert _{{\mathcal {M}}_r}+\Vert u_h^{n}\Vert _{{\mathcal {M}}_r}),$ we have:

$$\begin{aligned}&\left\| u_h^{m}\right\| ^2_{{\mathcal {M}}_r} +C\tau \sum \limits _{n=0}^{m-1}\Vert \nabla u_h^{n+\frac{1}{2}}\Vert ^2 \le C\left( \left\| u_h^{0}\right\| ^2_{{\mathcal {M}}_r}\right. \nonumber \\&\quad \left. +\tau ^{1-\alpha }\sum \limits _{n=0}^{m-1}|\lambda _{n+1}^{(\alpha )}|\left\| u_h^{0}\right\| ^2_{{\mathcal {M}}_r} +\tau \sum \limits _{n=0}^{m-1}\Vert f^{n+\frac{1}{2}}\Vert ^2_{{\mathcal {M}}_r}\right) +C\tau \sum \limits _{n=0}^{m-1}\Vert u_h^{n}\Vert ^2. \end{aligned}$$

(3.13)

Using the discrete Gronwall lemma, (2.3), and (3.13), we complete the proof of Theorem 3.2. $\square$

3.2 $H^1$ stability analysis

In the following theorem, we derive the $H^1$ stability of the Crank–Nicolson OSC scheme (2.6).

Theorem 3.3

The Crank–Nicolson OSC scheme (2.6) is stable with respect to $H^1$ norm. Specifically, for $u^m_h\in {\mathcal {M}}_r(\delta )$, $\ 1\le m\le K$, it holds:

$$\begin{aligned} \Vert \nabla u_h^{m}\Vert ^2 \le C\left( \Vert \nabla u_h^{0}\Vert ^2+\tau ^{1-\alpha }\Vert u_h^{0}\Vert ^2_{{\mathcal {M}}_r} +\tau \sum \limits _{n=0}^{m-1}\Vert f^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r}^2\right) . \end{aligned}$$

(3.14)

Proof

Setting $v_h=\delta _tu_h^{n+1}$ in (2.6), for $0\le n\le K-1$, we obtain:

$$\begin{aligned}&\Vert \delta _t u_h^{n+1}\Vert ^2_{{\mathcal {M}}_r} +\left\langle {\mathcal {L}}^{n+\frac{1}{2}}u_h^{n+\frac{1}{2}},\delta _tu_h^{n+1}\right\rangle \nonumber \\&\quad =-\tau ^{-\alpha }\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\left\langle u_h^{n-k+\frac{1}{2}},\delta _tu_h^{n+1}\right\rangle \nonumber \\&\qquad -\frac{\tau ^{-\alpha }}{2}\lambda _{n+1}^{(\alpha )}\left\langle u_h^{0},\delta _tu_h^{n+1}\right\rangle +\left\langle f^{n+\frac{1}{2}},\delta _tu_h^{n+1}\right\rangle . \end{aligned}$$

(3.15)

First, we have to handle the first term on RHS of (3.15) as follows:

$$\begin{aligned}&-\tau ^{-\alpha }\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\left\langle u_h^{n-k+\frac{1}{2}},\delta _tu_h^{n+1}\right\rangle \nonumber \\&\quad = -\tau ^{-\alpha }\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\left\langle \frac{u_h^{n-k+1}+u_h^{n-k}}{2},\frac{u_h^{n+1}-u_h^{n}}{\tau }\right\rangle \nonumber \\&\quad =-\frac{\tau ^{1-\alpha }}{2}\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\left\langle \delta _tu_h^{n-k+1},\delta _tu_h^{n+1}\right\rangle -\tau ^{-\alpha }\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\left\langle u_h^{n-k},\delta _tu_h^{n+1}\right\rangle . \end{aligned}$$

(3.16)

Now, we handle the second term on LHS of (3.15), following (3.1) of Lemma 3.1, we have:

$$\begin{aligned}&\left\langle {\mathcal {L}}^{n+\frac{1}{2}}u_h^{n+\frac{1}{2}},\delta _tu_h^{n+1}\right\rangle \nonumber \\&\quad =A_0(t_{n+\frac{1}{2}};u_h^{n+\frac{1}{2}},\delta _tu_h^{n+1}) +A_1(t_{n+\frac{1}{2}};u_h^{n+\frac{1}{2}},\delta _tu_h^{n+1}), \end{aligned}$$

(3.17)

since

$$\begin{aligned}&A_0(t_{n+\frac{1}{2}};u_h^{n+\frac{1}{2}},\delta _tu_h^{n+1}) \nonumber \\&\quad =\frac{1}{2\tau }A_0(t_{n+\frac{1}{2}};u_h^{n+1}+u_h^{n},u_h^{n+1}-u_h^{n}) \nonumber \\&\quad =\frac{1}{2\tau }\left[ A_0(t_{n+\frac{1}{2}};u_h^{n+1},u_h^{n+1})-A_0(t_{n-\frac{1}{2}};u_h^{n},u_h^{n})\right] \nonumber \\&\quad -\frac{1}{2\tau }\left[ A_0(t_{n+\frac{1}{2}};u_h^{n},u_h^{n})-A_0(t_{n-\frac{1}{2}};u_h^{n},u_h^{n})\right] \nonumber \\&\quad =\frac{1}{2}\delta _t\left( A_0(t_{n+\frac{1}{2}};u_h^{n+1},u_h^{n+1})\right) -\frac{1}{2}(\delta _tA_0)(t_{n+\frac{1}{2}};u_h^{n},u_h^{n}); \end{aligned}$$

(3.18)

substituting (3.16)–(3.18) into (3.15), multiplying the result equation by $2\tau$, and then summing from $n=1$ to $n=m-1$, $1\le m\le K$, we obtain:

$$\begin{aligned}&2\tau \sum \limits _{n=1}^{m-1}\Vert \delta _t u_h^{n+1}\Vert ^2_{{\mathcal {M}}_r} +A_0(t_{m-\frac{1}{2}};u_h^{m},u_h^{m}) = A_0(t_{\frac{1}{2}};u_h^{1},u_h^{1}) \nonumber \\&\quad +\,\tau \sum \limits _{n=1}^{m-1}(\delta _tA_0)(t_{n+\frac{1}{2}};u_h^{n},u_h^{n}) -2\tau \sum \limits _{n=1}^{m-1}A_1(t_{n+\frac{1}{2}};u_h^{n+\frac{1}{2}},\delta _tu_h^{n+1}) \nonumber \\&\quad -\,\tau ^{2-\alpha }\sum \limits _{n=1}^{m-1}\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\langle \delta _tu_h^{n-k+1},\delta _tu_h^{n+1}\rangle \nonumber \\&\quad -\,2\tau ^{1-\alpha }\sum \limits _{n=1}^{m-1} \sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\langle u_h^{n-k},\delta _tu_h^{n+1}\rangle \nonumber \\&\quad -\,\tau ^{1-\alpha }\sum \limits _{n=1}^{m-1}\lambda _{n+1}^{(\alpha )}\langle u_h^{0},\delta _tu_h^{n+1}\rangle +2\tau \sum \limits _{n=1}^{m-1}\langle f^{n+\frac{1}{2}},\delta _t u_h^{n+1}\rangle . \end{aligned}$$

(3.19)

Taking $n=0$ in (3.15), we obtain:

$$\begin{aligned}&\Vert \delta _t u_h^{1}\Vert ^2_{{\mathcal {M}}_r} +\langle {\mathcal {L}}^{\frac{1}{2}}u_h^{\frac{1}{2}},\delta _tu_h^{1}\rangle =-\tau ^{-\alpha }\lambda _0^{(\alpha )}\langle u_h^{1},\delta _tu_h^{1}\rangle \nonumber \\&\quad -\frac{\tau ^{-\alpha }}{2}\lambda _{1}^{(\alpha )}\langle u_h^{0},\delta _tu_h^{1}\rangle +\langle f^{\frac{1}{2}},\delta _tu_h^{1}\rangle , \ \ 0\le n\le K-1; \end{aligned}$$

(3.20)

following (3.1):

$$\begin{aligned} \langle {\mathcal {L}}^{\frac{1}{2}}u_h^{\frac{1}{2}},\delta _tu_h^{1}\rangle= & {} \frac{1}{2\tau }\langle {\mathcal {L}}^{\frac{1}{2}}(u_h^{1}+u_h^{0}),u_h^{1}-u_h^{0}\rangle \nonumber \\= & {} \frac{1}{2\tau }\left[ A_0(t_{\frac{1}{2}};u_h^{1}+u_h^{0},u_h^{1}-u_h^{0}) +A_1(t_{\frac{1}{2}};u_h^{1}+u_h^{0},u_h^{1}-u_h^{0})\right] \nonumber \\= & {} \frac{1}{2\tau }\left[ A_0(t_{\frac{1}{2}};u_h^{1},u_h^{1}) -A_0(t_{\frac{1}{2}};u_h^{0},u_h^{0})\right] + \frac{1}{2\tau }A_1(t_{\frac{1}{2}};u_h^{1}+u_h^{0},u_h^{1}-u_h^{0}).\nonumber \\ \end{aligned}$$

(3.21)

Furthermore, substituting (3.21) into (3.20), multiplying the resulting expression by $2\tau$, we have:

$$\begin{aligned}&2\tau \Vert \delta _t u_h^{1}\Vert ^2_{{\mathcal {M}}_r} +A_0(t_{\frac{1}{2}};u_h^{1},u_h^{1}) \nonumber \\&\quad =A_0(t_{\frac{1}{2}};u_h^{0},u_h^{0})-A_1(t_{\frac{1}{2}};u_h^{1}+u_h^{0},u_h^{1}-u_h^{0}) \nonumber \\&\qquad -2\tau ^{1-\alpha }\lambda _0^{(\alpha )}\langle u_h^{1},\delta _tu_h^{1}\rangle -\tau ^{1-\alpha }\lambda _{1}^{(\alpha )}\langle u_h^{0},\delta _tu_h^{1}\rangle +2\tau \langle f^{\frac{1}{2}},\delta _tu_h^{1}\rangle ; \end{aligned}$$

(3.22)

adding (3.22)–(3.19), for $1\le m\le K$, we have:

$$\begin{aligned}&2\tau \sum \limits _{n=0}^{m-1}\Vert \delta _t u_h^{n+1}\Vert ^2_{{\mathcal {M}}_r} +A_0(t_{m-\frac{1}{2}};u_h^{m},u_h^{m}) \nonumber \\&\quad = A_0(t_{\frac{1}{2}};u_h^{0},u_h^{0}) +\tau \sum \limits _{n=1}^{m-1}(\delta _tA_0)(t_{n+\frac{1}{2}};u_h^{n},u_h^{n}) \nonumber \\&\quad -\,2\tau \sum \limits _{n=0}^{m-1}A_1(t_{n+\frac{1}{2}};u_h^{n+\frac{1}{2}},\delta _tu_h^{n+1}) \nonumber \\&\quad -\tau ^{2-\alpha }\sum \limits _{n=1}^{m-1}\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\langle \delta _tu_h^{n-k+1},\delta _tu_h^{n+1}\rangle \nonumber \\&\quad -\,2\tau ^{1-\alpha }\sum \limits _{n=0}^{m-1} \sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\langle u_h^{n-k},\delta _tu_h^{n+1}\rangle \nonumber \\&\quad -\,\tau ^{1-\alpha }\sum \limits _{n=0}^{m-1}\lambda _{n+1}^{(\alpha )}\langle u_h^{0},\delta _tu_h^{n+1}\rangle +2\tau \sum \limits _{n=0}^{m-1}\langle f^{n+\frac{1}{2}},\delta _t u_h^{n+1}\rangle . \end{aligned}$$

(3.23)

It follows from (2.4) that:

$$\begin{aligned} -\tau ^{2-\alpha }\sum \limits _{n=1}^{m-1}\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\langle \delta _tu_h^{n-k+1},\delta _tu_h^{n+1}\rangle \le 0; \end{aligned}$$

(3.24)

using (3.10) and (3.3), we have:

$$\begin{aligned} A_0(t_{m-\frac{1}{2}};u_h^{m},u_h^{m})\ge p_{min} \Vert \nabla u_h^{m}\Vert ^2,\ \ A_0(t_{\frac{1}{2}};u_h^{0},u_h^{0})\le p_{max} \Vert \nabla u_h^{0}\Vert ^2; \end{aligned}$$

(3.25)

using Eq. (3.5) of Fernandes and Fairweather (1993), we have:

$$\begin{aligned} \left| \left\langle -\Delta u_h^{n},u_h^{n}\right\rangle \right| \; \le \; C\left\| \nabla u_h^{n}\right\| \left\| \nabla u_h^{n}\right\| ; \end{aligned}$$

(3.26)

also, using (3.26) and (3.4) in Lemma 3.1, we have:

$$\begin{aligned} (\delta _tA_0)(t_{n+\frac{1}{2}};u_h^{n},u_h^{n})\le & {} \frac{1}{\tau }\left| A_0(t_{n+\frac{1}{2}};u_h^{n},u_h^{n})-A_0(t_{n-\frac{1}{2}};u_h^{n},u_h^{n})\right| \nonumber \\\le & {} C \langle -\Delta u_h^{n},u_h^{n}\rangle \le C_1 \Vert \nabla u_h^{n}\Vert ^2. \end{aligned}$$

(3.27)

Similarly, using (3.5) in Lemma 3.1 and (3.26), we have:

$$\begin{aligned} A_1(t_{n+\frac{1}{2}};u_h^{n+\frac{1}{2}},\delta _tu_h^{n+1})\le C \Vert \nabla u_h^{n+\frac{1}{2}}\Vert \ \Vert \delta _tu_h^{n+1}\Vert _{{\mathcal {M}}_r}. \end{aligned}$$

(3.28)

Thus, on substituting (3.24)–(3.28) into (3.23), dropping the non-positive the fourth term on the RHS of (3.23), and then applying the Cauchy–Schwarz inequality to the last three terms on the RHS of the resulting expression, we have:

$$\begin{aligned}&2\tau \sum \limits _{n=0}^{m-1}\Vert \delta _t u_h^{n+1}\Vert ^2_{{\mathcal {M}}_r} + p_{\text {min}} \Vert \nabla u_h^{m}\Vert ^2 \nonumber \\&\quad \le p_{\text {max}} \Vert \nabla u_h^{0}\Vert ^2 +C\tau \sum \limits _{n=1}^{m-1} \Vert \nabla u_h^{n}\Vert ^2 +2\tau C\sum \limits _{n=0}^{m-1} \Vert \nabla u_h^{n+\frac{1}{2}}\Vert \ \Vert \delta _tu_h^{n+1}\Vert _{{\mathcal {M}}_r} \nonumber \\&\qquad +2\tau ^{1-\alpha }\sum \limits _{n=0}^{m-1} \sum \limits _{k=0}^{n}|\lambda _k^{(\alpha )}|\ \Vert u_h^{n-k}\Vert _{{\mathcal {M}}_r}\ \Vert \delta _tu_h^{n+1}\Vert _{{\mathcal {M}}_r} \nonumber \\&\qquad +\tau ^{1-\alpha }\sum \limits _{n=0}^{m-1}|\lambda _{n+1}^{(\alpha )}|\ \Vert u_h^{0}\Vert _{{\mathcal {M}}_r}\ \Vert \delta _tu_h^{n+1}\Vert _{{\mathcal {M}}_r} \nonumber \\&\qquad +2\tau \sum \limits _{n=0}^{m-1}\Vert f^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r}\ \Vert \delta _t u_h^{n+1}\Vert _{{\mathcal {M}}_r}. \end{aligned}$$

(3.29)

Using (2.3), noticing $\Vert \nabla u_h^{n+\frac{1}{2}}\Vert \le \frac{1}{2}(\Vert \nabla u_h^{n+1}\Vert +\Vert \nabla u_h^{n}\Vert )$, $\Vert u_h^{n-k}\Vert _{{\mathcal {M}}_r}\le C\Vert \nabla u_h^{n-k}\Vert _{{\mathcal {M}}_r}$, applying the Young’s inequality, and simplifying, we have:

$$\begin{aligned}&\tau \sum \limits _{n=0}^{m-1}\Vert \delta _t u_h^{n+1}\Vert ^2_{{\mathcal {M}}_r} + (p_{\text {min}}-C\tau ) \Vert \nabla u_h^{m}\Vert ^2 \nonumber \\&\quad \le C\left( \Vert \nabla u_h^{0}\Vert ^2+\tau ^{1-\alpha }\Vert u_h^{0}\Vert ^2_{{\mathcal {M}}_r} +2\tau \sum \limits _{n=0}^{m-1}\Vert f^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r}^2\right) +C\tau \sum \limits _{n=1}^{m-1} \Vert \nabla u_h^{n}\Vert ^2;\qquad \end{aligned}$$

(3.30)

by choosing $\tau$ small so that $p_{\text {min}}-C\tau >0$, and using the discrete Gronwall lemma, we have:

$$\begin{aligned} \Vert \nabla u_h^{m}\Vert ^2 \le C\left( \Vert \nabla u_h^{0}\Vert ^2+\tau ^{1-\alpha }\Vert u_h^{0}\Vert ^2_{{\mathcal {M}}_r} +2\tau \sum \limits _{n=0}^{m-1}\Vert f^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r}^2\right) ,\ 1\le m\le K. \end{aligned}$$

The proof of Theorem 3.3 is completed. $\square$

3.3 Convergence and a superconvergence result

In this subsection, we will consider the convergence of Crank–Nicolson OSC scheme. To analyze the convergence, we need to define an elliptic projection W: $[0,T]\rightarrow {\mathcal {M}}_r(\delta )$, for $\ t\in [0,T]$:

$$\begin{aligned} \langle {\mathcal {L}}u-{\mathcal {L}}W,\nu _h\rangle =0,\ \ {\forall \ \nu _h\in {\mathcal {M}}_r(\delta )}. \end{aligned}$$

(3.31)

As in Bialecki (1998), for a given function u, Eq. (3.31) has a unique solution $W\in {\mathcal {M}}_r(\delta )$.

To finish our analysis, we now introduce two lemmas which provide estimates for $u-W$ and its time derivatives.

Lemma 3.4

Bialecki (1998) Assume u, $\partial u/\partial t \in H^{r+3-j},\ j=0, 1$, and W satisfies (3.31), and then, there exists a constant C, independent of h, such that:

$$\begin{aligned} \left\| \frac{{\partial }^i(u-W)}{\partial t^i}\right\| _{H^j}\le Ch^{r+1-j}\left\| \frac{{\partial }^iu}{\partial t^i}\right\| _{H^{r+3-j}},\quad j=0, 1,\ \ i=0, 1. \end{aligned}$$

(3.32)

Lemma 3.5

Bialecki (1998) Assume u, $\partial u/\partial t\in H^{r+3}$, for $t\in \left[ 0,T\right]$, $l=l_1+l_2$, we have:

$$\begin{aligned} \left\| \frac{\partial ^{l+i}(u-W)}{\partial x^{l_1}\partial y^{l_2}\partial t^i}\right\| _{{\mathcal {M}}_r}\le Ch^{r+1-l}\left\| \frac{{\partial }^iu}{\partial t^i}\right\| _{H^{r+3}},\quad i=0, 1,\ 0\le l\le 4. \end{aligned}$$

(3.33)

Now, we derive an optimal $H^{\ell }$ ($\ell =0,1$) error estimate.

According to the definition of W in (3.31), for $0 \le n \le K$, we assume:

$$\begin{aligned} \zeta ^{n}=-W^{n}+u_h^n, \quad \eta ^{n}=-W^{n}+u^{n}, \end{aligned}$$

(3.34)

then

$$\begin{aligned} u^n-u_h^{n}=W^{n}+\eta ^{n}-(\zeta ^{n}+W^{n})=\eta ^{n}-\zeta ^{n}. \end{aligned}$$

(3.35)

It is easy to known that the estimates of $\eta ^{n}$ are known from Lemmas 3.4 and 3.5. Therefore, to bound $u^n-u_h^{n}$, we need only to bound $\zeta ^{n}$.

First, from (1.1) and (3.31) at $t=t_{n+\frac{1}{2}}$, (2.6), (3.34), and (3.35), and then for $v_h\in {\mathcal {M}}_r(\delta )$, we obtain:

$$\begin{aligned}&\langle \delta _t\zeta ^{n+1},v_h\rangle +\langle {\mathcal {L}}^{n+\frac{1}{2}}\zeta ^{n+\frac{1}{2}},v_h\rangle =-\tau ^{-\alpha }\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )}\langle \zeta ^{n-k+\frac{1}{2}},v_h\rangle \nonumber \\&\quad -\frac{\tau ^{-\alpha }}{2}\lambda _{n+1}^{(\alpha )}\langle \zeta ^{0},v_h\rangle +\langle \sigma _{\alpha ,u}^{n+\frac{1}{2}},v_h\rangle , \quad v_h\in {\mathcal {M}}_r(\delta ),\ 0\le n\le K-1, \end{aligned}$$

(3.36)

where

$$\begin{aligned} \sigma ^{n+\frac{1}{2}}_{\alpha ,u}=\sigma ^{n+\frac{1}{2}}_1+\sigma ^{n+\frac{1}{2}}_2 +\sigma ^{n+\frac{1}{2}}_3+\sigma ^{n+\frac{1}{2}}_4+\sigma ^{n+\frac{1}{2}}_5, \end{aligned}$$

(3.37)

and

$$\begin{aligned} \left\{ \begin{array}{ll} \sigma _1^{n+\frac{1}{2}}=\delta _t\eta ^{n+1}; \\ \sigma _2^{n+\frac{1}{2}}=u_t(t_{n+\frac{1}{2}})-\delta _t u^{n+1}; \\ \sigma _3^{n+\frac{1}{2}}=L^{n+\frac{1}{2}}\left( W\left( t_{n+\frac{1}{2}}\right) -W^{n+\frac{1}{2}}\right) ; \\ \sigma _4^{n+\frac{1}{2}}=\tau ^{-\alpha }\sum \limits _{k=0}^{n}\lambda _k^{(\alpha )} \eta ^{n-k+\frac{1}{2}}+\frac{\tau ^{-\alpha }}{2}\lambda _{n+1}^{(\alpha )} \eta ^{0}; \\ \sigma _5^{n+\frac{1}{2}}=R_{(\alpha )}^{n+\frac{1}{2}}. \end{array} \right. \end{aligned}$$

In the following lemma, we derive estimates on $\sigma _{\alpha ,u}^{n+\frac{1}{2}}$ that are required to prove the convergence estimates for the proposed Crank–Nicolson OSC scheme in $H^{\ell }$ ($\ell =0,1$) norms on each time level.

Lemma 3.6

If $u\in C^{2,0,0}\cap C^{0,2,0} \cap C^{0,0,3}$, $^{RL}_0D^{\alpha }_tu, u_{tt}\in C\left( [0,T],H^{r+3}\right)$, $^{RL}_0D^{\alpha +2}_tu$ and its Fourier transform belong to $L^{1}({\mathbb {R}})$, for $n=0,1,\ldots ,K-1$, then we have:

$$\begin{aligned} \left\| \sigma _{\alpha ,u}^{n+\frac{1}{2}}\right\| _{{\mathcal {M}}_r}\le & {} C h^{r+1}\left( \left\| u_t\right\| _{C([0,T],H^{r+3})} +\left\| ^{RL}_0D^{\alpha }_tu\right\| _{C([0,T],H^{r+3})}\right) \nonumber \\&+\,C\tau ^2\left( \left\| u_{ttt}\right\| _{C(\Omega _T)}+\Vert u_{tt}\Vert _{C^{2,0,0}\cap C^{0,2,0}} \right. \nonumber \\&+\left. \left\| u_{tt}\right\| _{C([0,T],H^{r+3})} +\left\| {\mathfrak {F}}[^{RL}_0D^{\alpha +2}_tu ](\omega )\right\| _{L^1} \right) . \end{aligned}$$

(3.38)

Proof

Since

$$\begin{aligned} \left\| \sigma _1^{n+\frac{1}{2}}\right\| _{{\mathcal {M}}_r}= & {} \frac{1}{\tau }\left\| \int ^{t_{n+1}}_{t_n}\frac{\partial \eta }{\partial t}(\cdot ,s){\text {d}}s\right\| _{{\mathcal {M}}_r} \nonumber \\\le & {} \frac{1}{\tau }\int ^{t_{n+1}}_{t_n}\left\| \frac{\partial \eta }{\partial t}(\cdot ,s)\right\| _{{\mathcal {M}}_r} {\text {d}}s \le Ch^{r+1}\left\| u_t\right\| _{C\left( [0,T],H^{r+3}\right) }. \end{aligned}$$

(3.39)

Using Taylor’s theorem with integral remainder, we obtain:

$$\begin{aligned} \Vert \sigma _2^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r}=\Vert u_t(t_{n+\frac{1}{2}})-\delta _t u^{n+1}\Vert _{{\mathcal {M}}_r} \le C\tau ^2\left\| u_{ttt}\right\| _{C(\Omega _T)}. \end{aligned}$$

(3.40)

For the term $\sigma _3^{n+\frac{1}{2}}$, we obtain, on using Taylor’s theorem and the boundedness of the coefficients:

$$\begin{aligned} \Vert \sigma _3^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r}= & {} \Vert L^{n+\frac{1}{2}} \left( W\left( t_{n+\frac{1}{2}}\right) -W^{n+\frac{1}{2}}\right) \Vert _{{\mathcal {M}}_r} \nonumber \\\le & {} C\tau ^2\left\| W(t_{n+\frac{1}{2}})-W^{n+\frac{1}{2}} \right\| _{C^{2,0,0}\cap C^{0,2,0}} \nonumber \\\le & {} C \tau ^2 \Vert W_{tt}\Vert _{C^{2,0,0}\cap C^{0,2,0}}. \end{aligned}$$

(3.41)

Since

$$\begin{aligned} \Vert W_{tt}\Vert _{C^{2,0,0}\cap C^{0,2,0}}\le \Vert (W-u)_{tt}\Vert _{C^{2,0,0}\cap C^{0,2,0}}+\Vert u_{tt}\Vert _{C^{2,0,0}\cap C^{0,2,0}}, \end{aligned}$$

(3.42)

then, using Taylor’s theorem and arguments as in (3.39), together with Lemma 3.5 ($l=2$, $j=2$), since $r\ge 3$, it follows on using (Fernandes and Fairweather 1993, Lemma 3.2) that:

$$\begin{aligned}&\Vert \sigma _3^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r} \nonumber \\&\quad \le C\tau ^2\left( \ \left\| (W-u)_{tt}\right\| _{C^{2,0,0}\cap C^{0,2,0}}+\Vert u_{tt}\Vert _{C^{2,0,0}\cap C^{0,2,0}}\right) \nonumber \\&\quad \le C\tau ^2\left( \ \Vert u_{tt}\Vert _{C^{2,0,0}\cap C^{0,2,0}}+ \left\| u_{tt}\right\| _{C([0,T],H^{r+3})}\right) . \end{aligned}$$

(3.43)

By (2.5), we know that:

$$\begin{aligned} \left\| ^{RL}_0D^{\alpha }_t\eta (t_{n+\frac{1}{2}})-\sigma _4^{n+\frac{1}{2}}\right\| _{{\mathcal {M}}_r} \le C\tau ^2\left\| {\mathfrak {F}}[^{RL}_0D^{\alpha +2}_tu](\omega )\right\| _{L^1}. \end{aligned}$$

Hence, using Lemma 3.5, we have:

$$\begin{aligned} \left\| \sigma _4^{n+\frac{1}{2}}\right\| ^2_{{\mathcal {M}}_r}\le & {} \left\| ^{RL}_0D^{\alpha }_t\eta (t_{n+\frac{1}{2}})\right\| ^2_{{\mathcal {M}}_r} +\tau ^4\left\| {\mathfrak {F}}[^{RL}_0D^{\alpha +2}_tu](\omega )\right\| ^2_{L^1} \nonumber \\\le & {} Ch^{2r+2}\left\| ^{RL}_0D^{\alpha }_tu\right\| ^2_{C([0,T],H^{r+3})} +\tau ^4\left\| {\mathfrak {F}}[^{RL}_0D^{\alpha +2}_tu](\omega )\right\| ^2_{L^1}. \end{aligned}$$

(3.44)

Note that

$$\begin{aligned} \left\| \sigma _5^{n+\frac{1}{2}}\right\| _{{\mathcal {M}}_r} =\left\| R_{(\alpha )}^{n+\frac{1}{2}}\right\| _{{\mathcal {M}}_r} \le C\tau ^2\left\| {\mathfrak {F}}[^{RL}_0D^{\alpha +2}_tu ](\omega )\right\| _{L^1}. \end{aligned}$$

(3.45)

Applying the triangle inequality to (3.37) and using (3.39)–(3.40) and (3.43)–(3.45) yield (3.38). The proof of the Lemma 3.6 is completed. $\square$

Convergence estimates for the Crank–Nicolson OSC method (2.6) in the $H^{\ell }$ norms, $\ell =0,1,$ are proved in the following theorem.

Theorem 3.7

If the hypotheses of Lemma 3.6 are satisfied and suppose that u is the solution of (1.1), and $u_h^m$ ($0\le m\le K$) is the solution of the problem (2.6) with $u^0_h=W^0$, then there exists a positive constant C, independent of h and $\tau$, such that

$$\begin{aligned} \left\| u(t_{m})-u_{h}^{m}\right\| _{H^{j}}\le C\left( h^{r+1-j}+{\tau }^{2}\right) , \quad j =0, 1, \quad 0\le m\le K. \end{aligned}$$

(3.46)

Proof

We first apply the stability result (3.6) and (3.14)–(3.13) to obtain:

$$\begin{aligned} \left\| \zeta ^{m}\right\| ^2_{{\mathcal {M}}_r} \le C\left( \tau ^{1-\alpha }\left\| \zeta ^{0}\right\| ^2_{{\mathcal {M}}_r} +\tau \sum \limits _{n=0}^{m-1}\Vert \sigma _{\alpha , u}^{n+\frac{1}{2}}\Vert ^2_{{\mathcal {M}}_r}\right) , \quad 1\le m\le K, \end{aligned}$$

(3.47)

and

$$\begin{aligned} \Vert \nabla \zeta ^{m}\Vert ^2 \le C\left( \Vert \nabla \zeta ^{0}\Vert ^2+\tau ^{1-\alpha }\Vert \zeta ^{0}\Vert ^2_{{\mathcal {M}}_r} +\tau \sum \limits _{n=0}^{m-1}\Vert \sigma _{\alpha , u}^{n+\frac{1}{2}}\Vert _{{\mathcal {M}}_r}^2\right) . \end{aligned}$$

(3.48)

Since $\zeta ^0=0$, it follows from Lemma 3.6 that:

$$\begin{aligned} \left\| \zeta ^{m}\right\| _{{\mathcal {M}}_r} \le C\left( h^{r+1}+\tau ^{2}\right) , \quad 1\le m\le K, \end{aligned}$$

(3.49)

and

$$\begin{aligned} \Vert \nabla \zeta ^{m}\Vert \le C\left( h^{r+1}+\tau ^{2}\right) ,\quad 1\le m\le K. \end{aligned}$$

(3.50)

Therefore, (3.46) for $j=0$ and $j=1$ is obtained on using the triangle inequality, (3.49) and (3.50), respectively, and (3.32) with $l=0, j=0$ and $l=0, j=1$, respectively. $\square$

Remark 3.8

If we choose $u^0_h$ as the elliptic projection $W^0$ of $u_0$ defined in (3.31), then $\zeta ^0=0$. Hence, from (3.50), we obtain a superconvergence result for $\Vert \zeta ^{m}\Vert _{H^1}$, $1\le m\le K$, namely:

$$\begin{aligned} \Vert \zeta ^{m}\Vert _{H^1} \le C\left( h^{r+1}+\tau ^{2}\right) ,\quad 1\le m\le K. \end{aligned}$$

(3.51)

From Sobolev’s inequality, we obtain, since: $\zeta ^{m}\in {\mathcal {M}}_r$,

$$\begin{aligned} \Vert \zeta ^{m}\Vert _{L^{\infty }}\le C \log \left( \frac{1}{h}\right) \quad | \nabla \zeta ^{m}\Vert , 1\le m\le K. \end{aligned}$$

(3.52)

If the optimal maximum norm estimate for $\eta ^m$ are available, namely:

$$\begin{aligned} \Vert \eta ^{m}\Vert _{L^{\infty }}\le C h^{r+1},\quad 1\le m\le K. \end{aligned}$$

(3.53)

Then, on using the triangle inequality, we obtain a quasi-optimal $L^{\infty }$ error estimate:

$$\begin{aligned} \left\| u(t_{m})-u_h^{m}\right\| _{L^{\infty }}\le C\log \left( \frac{1}{h}\right) \left( h^{r+1}+\tau ^{2}\right) , \quad 1\le m\le K. \end{aligned}$$

(3.54)

Remark 3.9

If the hypotheses of Lemma 3.6 are satisfied, then (3.46) also holds for $j=0$ and $j=1$, and suppose $u_h^0$ is chosen, so that

$$\begin{aligned} \left\| u_0-u_h^{0}\right\| _{H^j}\le Ch^{r+1-j}, \quad j =0, 1. \end{aligned}$$

(3.55)

This is satisfied by the choice $u_h^0=u_{{\mathcal {H}}}^0$, the Hermite interpolant of $u_0$ defined in Bialecki (Eq. 2.18, Bialecki 1998).

4 Numerical experiments

In this section, we will present numerical experiments to illustrate our theoretical statements. We used the space of piecewise Hermite bicubics ($r=3$) with the standard value and scaled slope basis functions Yan and Fairweather (1992) on uniform partitions of [0, 1].

Example 1

We consider the following problem similar to Chen et al. (2016):

$$\begin{aligned} \left\{ \begin{array}{ll} u_t+^C_0D^{\alpha }_tu-(2-\sin (tx))u_{xx}+t\cos (tx)u_x+(2-\cos (tx))u=f(x,t), \\ u(x,0)=0,\quad x\in [0,1], \\ u(x,t)=0, \quad t\in (0,1], \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} f(x,t)= & {} \left( \frac{2}{\Gamma (3-\alpha )}t^{2-\alpha }+2t+(2-\cos (tx))t^2\right) \frac{\sin (2\pi x)}{(1+x)^2}\\&\quad +\,t^3\cos (tx)\left( \frac{2\pi \cos (2\pi x)}{(1+x)^2}-\frac{2\sin (2\pi x)}{(1+x)^3}\right) \\&\quad +t\,^2(2-\sin (tx))\left( 4\pi ^2\frac{\sin (2\pi x)}{(1+x)^2}+\frac{8\pi \cos (2\pi x)}{(1+x)^3}-\frac{6\sin (2\pi x)}{(1+x)^4}\right) , \end{aligned}$$

with the exact solution $u(x,t)=t^2\frac{\sin (2\pi x)}{(1+x)^2}.$

Table 1 $L^2$ and $L^{\infty }$ errors and convergence rates in spatial and temporal directions with $\tau =h^2$ for Example 1

Table 2 $L^2$ and $L^{\infty }$ errors and convergence rates in temporal direction with $\tau =h$ for Example 1

Table 3 $H^1$ errors and convergence rates with $\tau =h^{\frac{3}{2}}$ for Example 1

In Table 1, we select $\tau =h^2$ $(K=N^2)$, since, from our theoretical estimates, the error in the $L^2$ norm is expected to be ${\mathcal {O}}(\tau ^{2}+h^{4})$ when $r=3$. Just as we hope, the results in Table 1 demonstrate the expected convergence rates of 4 order in space and 2 in time for different $\alpha$ ($\alpha =0.15,0.5,0.95$).

We now verify the temporal accuracy and convergence rates for our proposed method, and select $\tau =h$ $(K=N)$, so that the error stemming from the spatial approximation is negligible. Table 2 verifies 2 order accuracy in time for all four different $\alpha$ ($\alpha =0.1,0.5,0.99$), which are in keeping with the theoretical predictions.

By selecting $\tau =h^{3/2}$ and different $\alpha$ ($\alpha =0.01,0.4,0.7,0.9$), Table 3 indicates $H^1$ errors and convergence rates in spatial direction. The convergence rate of 3 order matches that of the theoretical one.

Example 2

We consider the following problem similar to Chen et al. (2016):

$$\begin{aligned} \left\{ {\begin{array}{*{20}l} {u_{t} + _{0}^{C} D_{t}^{\alpha } u - (2 - \sin (tx))u_{{xx}} + t\cos (tx)u_{x} + (2 - \cos (tx))u = f(x,t),} \\ {u(x,0) = 0,\quad x \in \Omega ,} \\ {u(x,t) = 0,\quad (x,t) \in \partial \Omega \times (0,T],} \\ \end{array} } \right. \end{aligned}$$

where $\Omega =[0,1]$, $T=1$:

$$\begin{aligned} f(x,t)= & {} \left( \Gamma (2+\alpha )t+(1+\alpha )t^{\alpha }+(2-\cos (tx))t^{1+\alpha }\right) x(1-x)e^{-x}\\&+t^3\cos (tx)(2-\sin (tx))(x^2-5x+4)e^{-x}t^{1+\alpha }\\&+t\cos (tx)(x^2-3x+1)e^{-x}t^{1+\alpha }. \end{aligned}$$

Table 4 $L^2$ and $L^{\infty }$ errors and convergence rates in spatial and temporal directions with $\tau =h^2$ for Example 2

Table 5 $L^2$ and $L^{\infty }$ errors and convergence rates in temporal direction with $\tau =h$ for Example 2

Table 6 $H^1$ errors and convergence rates with $\tau =h^{\frac{3}{2}}$ for Example 2

Tables 4, 5, 6 show the errors and convergence rates in three discrete norms for Example 2.

For the fractional order $\alpha =0.25,0.5,0.95$, Table 4 shows the $L^2$ and $L^{\infty }$ errors and convergence rates, and verifies that the space convergence rate is 4 and time convergence rate is 2 for each $\alpha$. It is obvious that the numerical convergence order matches well with the theoretical results.

In Table 5, for the fractional order $\alpha =0.01,0.35,0.65,0.99$, we present the convergence order in temporal direction. It is easy to conclude that the method is convergent and the convergence order in time is 2 corresponding to each $\alpha$.

We show the errors in $H^1$ norm for $\alpha =0.01,0.3,0.6,0.8$ in Table 6. It is clear that the convergence rate is three, which is the same as theoretically claimed.

Example 3

We consider the following problem Chen et al. (2016):

$$\begin{aligned} \left\{ {\begin{array}{*{20}l} {\mathop C\limits _{0} D_{t}^{\alpha } u - (2 - \sin (tx))u_{{xx}} + (2 - \cos (tx))u = f(x,t),} \\ {u(x,0) = 0,\quad x \in [0,1],} \\ {u(0,t) = u(1,t) = 0,\quad t \in (0,1],} \\ \end{array} } \right. \end{aligned}$$

with

$$\begin{aligned} f(x,t)= & {} \left( \Gamma (2+\alpha )t+(2-\cos (tx))t^{1+\alpha }\right) x(1-x)e^{-x} \\&\quad +t^3\cos (tx)(2-\sin (tx))(x^2-5x+4)e^{-x}t^{1+\alpha }. \end{aligned}$$

In this example, by choosing the same parameter h, $\tau$ and $\alpha$ as in Chen et al. (2016), we compare the numerical results of our scheme with the method in Chen et al. (2016). To eliminate the contamination of the spatial error, we choose $h=1/125$, which is large enough as the solution is analytic. Tables 7, 8 display $L^{2}$ and $L^{\infty }$ errors and the convergence orders with $\alpha = 0.1, 0.5, 0.9$, respectively. The last two columns of Tables 7 and 8 present the numerical results obtained in Chen et al. (2016). From Tables 7 and 8, we can see that the present method have similar accuracy and convergence order in time as reference Chen et al. (2016).

Table 7 Comparison of $L^{2}$ errors and convergence rate for Example 3 with $1/h=125$

Table 8 Comparison of $L^{\infty }$ errors and convergence rate for Example 3 with $1/h=125$

In the following Example 4, we mainly test problem based on the Gaussian pulse and the noise effect to show the efficiency of the developed technique.

Example 4

Let $\Omega =[0,1]$, $T=1$, we consider the following problem:

$$\begin{aligned} \left\{ {\begin{array}{*{20}l} {u_{t} + _{0}^{C} D_{t}^{\alpha } u - (2 - \sin (tx))u_{{xx}} + t\cos (tx)u_{x} + (2 - \cos (tx))u = f(x,t),} \\ {u(x,0) = 0,\quad x \in \Omega ,} \\ {u(x,t) = 0,\quad (x,t) \in \partial \Omega \times (0,T],} \\ \end{array} } \right. \end{aligned}$$

with the exact solution $u(x,t)=t^{2+\alpha }e^{-\frac{(x-0.5)^2}{\beta }}\sin (\pi x)$, where $\beta$ is small parameter.

Fig. 1

Left: the exact solution. Right: the numerical solution

Fig. 2

The absolute error of the numerical solution at $\alpha =0.5$ and $\tau ^2=h^4=\frac{1}{1600}$ for Example 4

In Fig. 1, we draw the surface figures of the exact solution u and the numerical solution $u_h$ with $h=1/40$, $\tau =1/1600$, $\alpha =0.5$, and $\beta =0.01$, respectively. We can clearly see that the exact solution u can be simulated well by the approximation solution $u_h$ for our discrete scheme in this case. In Fig. 2, we give the error surface figure for $|u-u_h|$. From the error figure, we can find that our numerical method can solve well the numerical solution in this case.

5 Conclusion

In the present work, we have developed an effective Crank–Nicolson OSC scheme for fractional-order mobile–immobile equation with variable coefficients. It is proved that our proposed fully methods are of optimal order in certain $H_j$ ($j=0,1$) norms. Also, $L^{\infty }$ estimates in space are derived. Some numerical examples have been carried out to verify the accuracy and efficiency of Crank–Nicolson OSC scheme.