Second-Order and Nonuniform Time-Stepping Schemes for Time Fractional Evolution Equations with Time–Space Dependent Coefficients

Abstract

The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time–space dependent variable coefficients is challenging, especially when the classical weak initial singularities are taken into account. In this paper, we introduce a concise technique to construct efficient time-stepping schemes with variable time step sizes for two-dimensional time fractional sub-diffusion and diffusion-wave equations with general time–space dependent variable coefficients. By means of the novel technique, the nonuniform Alikhanov type schemes are constructed and analyzed for the sub-diffusion and diffusion-wave problems. For the diffusion-wave problem, our scheme is constructed by employing the recently established symmetric fractional-order reduction method. The unconditional stability of proposed schemes is rigorously discussed under mild assumptions on variable coefficients and, based on reasonable regularity assumptions and weak time mesh restrictions, the second-order convergence is obtained with respect to discrete $H^1$-norm. Numerical experiments are given to demonstrate the theoretical statements.

Appendix

1.1 The Coefficients of Alikhanov Formulas

The coefficients $A_{n-k}^{(n)}$ of the Alikhanov formula on general meshes are defined as ([15])

$$\begin{aligned} A_{n-k}^{(n)}:=\left\{ \begin{array}{ll} a_0^{(n)}+\rho _{n-1}b_1^{(n)},\quad &{} k=n,\\ a_{n-k}^{(n)}+\rho _{k-1}b_{n-k+1}^{(n)}-b_{n-k}^{(n)}, &{} 2\le k\le n-1,\\ a_{n-1}^{(n)}-b_{n-1}^{(n)}, &{} k=1, \end{array}\right. \quad \text{ for }~n\ge 2, \end{aligned}$$

where

$$\begin{aligned}&a_{n-k}^{(n)}:=\frac{1}{\tau _k}\int _{t_{k-1}}^{\min \{t_k,t_{n-\theta }\}}\omega _{1-\beta }(t_{n-\theta }-s)\,\mathrm {d}s ,~1\le k\le n,\\&b_{n-k}^{(n)}:=\frac{2}{\tau _k(\tau _k+\tau _{k+1})}\int _{t_{k-1}}^{t_k}\omega _{1-\beta }(t_{n-\theta }-s)(s-t_{k-\frac{1}{2}})\,\mathrm {d}s ,~1\le k\le n-1, \end{aligned}$$

with $\rho _k:=\tau _k/\tau _{k+1}$ being the local time step-size ratios. It has been proved in [14, 15] that the discrete coefficients of the nonuniform Alikhanov formula (with $\pi _A=11/4$ and $\rho =7/4$, where $\rho :=\max _{k}\{\rho _k\}$ is the maximum step-size ratio) satisfy two basic properties:

A1.:

The discrete kernels are positive and monotone: $A_0^{(n)}\ge A_1^{(n)}\ge \cdots \ge A_{n-1}^{(n)}>0$;

A2.:

The discrete kernels fulfill $A_{n-k}^{(n)}\ge \frac{1}{\pi _A}\int _{t_{k-1}}^{t_k}\frac{\omega _{1-\beta }(t_n-s)}{\tau _k}\,\mathrm {d}s$ for $1\le k\le n\le N$.

1.2 The Proof of (2.1)

We will go through the proof of [14, Lemma A.1] to show that

$$\begin{aligned}&2(\mathbf{z}^{n})^T\mathbf{Q}^{(n)}({{\mathcal {D}}}_\tau ^\beta \mathbf{z})^{n-\theta }\ge \sum _{k=1}^nA_{n-k}^{(n)}\nabla _\tau [(\mathbf{z}^k)^T\mathbf{Q}^{(n)}{} \mathbf{z}^k]+\frac{(({{\mathcal {D}}}_\tau ^\beta \mathbf{z})^{n-\theta })^T\mathbf{Q}^{(n)}({{\mathcal {D}}}_\tau ^\beta \mathbf{z})^{n-\theta }}{A_0^{(n)}}, \end{aligned}$$

(7.1)

$$\begin{aligned}&2(\mathbf{z}^{n-1})^T\mathbf{Q}^{(n)}({{\mathcal {D}}}_\tau ^\beta \mathbf{z})^{n-\theta }\ge \sum _{k=1}^nA_{n-k}^{(n)}\nabla _\tau [(\mathbf{z}^k)^T\mathbf{Q}^{(n)}\mathbf{z}^k]-\frac{(({{\mathcal {D}}}_\tau ^\beta \mathbf{z})^{n-\theta })^T\mathbf{Q}^{(n)}({{\mathcal {D}}}_\tau ^\beta \mathbf{z})^{n-\theta }}{A_0^{(n)}-A_1^{(n)}}, \end{aligned}$$

(7.2)

for $1\le n\le N$ and $A_1^{(1)}:=0$.

For fix n, denote

$$\begin{aligned} J_n:=2(\mathbf{z}^{n})^T\mathbf{Q}^{(n)}({{\mathcal {D}}}_\tau ^\beta \mathbf{z})^{n-\theta }-\sum _{k=1}^nA_{n-k}^{(n)}\nabla _\tau [(\mathbf{z}^k)^T\mathbf{Q}^{(n)}{} \mathbf{z}^k]. \end{aligned}$$

Then

$$\begin{aligned} J_n&= \sum _{k=1}^nA_{n-k}^{(n)}\left[ 2(\mathbf{z}^{n})^T\mathbf{Q}^{(n)}(\mathbf{z}^{k}-\mathbf{z}^{k-1})-(\mathbf{z}^{k}+\mathbf{z}^{k-1})^T\mathbf{Q}^{(n)}(\mathbf{z}^{k}-\mathbf{z}^{k-1}) \right] \\&=\sum _{k=1}^nA_{n-k}^{(n)}\left[ \left( 2\mathbf{z}^{n}-(\mathbf{z}^{k}+\mathbf{z}^{k-1})\right) ^T\mathbf{Q}^{(n)}(\mathbf{z}^{k}-\mathbf{z}^{k-1}) \right] \\&=\sum _{k=1}^nA_{n-k}^{(n)}(\mathbf{z}^{k}-\mathbf{z}^{k-1})^T\mathbf{Q}^{(n)}(\mathbf{z}^{k}-\mathbf{z}^{k-1})\\&\quad +2\sum _{k=1}^nA_{n-k}^{(n)}\sum _{j=k+1}^n(\mathbf{z}^{j}-\mathbf{z}^{j-1})^T\mathbf{Q}^{(n)}(\mathbf{z}^{k}-\mathbf{z}^{k-1}) \\&=\sum _{k=1}^nA_{n-k}^{(n)}(\mathbf{z}^{k}-\mathbf{z}^{k-1})^T\mathbf{Q}^{(n)}(\mathbf{z}^{k}-\mathbf{z}^{k-1})\\&\quad +2\sum _{j=2}^n\sum _{k=1}^{j-1}A_{n-k}^{(n)}(\mathbf{z}^{j}-\mathbf{z}^{j-1})^T\mathbf{Q}^{(n)}(\mathbf{z}^{k}-\mathbf{z}^{k-1}). \end{aligned}$$

where the identity $2\mathbf{z}^{n}-(\mathbf{z}^{k}+\mathbf{z}^{k-1})=\mathbf{z}^{k}-\mathbf{z}^{k-1}+2\sum _{j=k+1}^n(\mathbf{z}^{j}-\mathbf{z}^{j-1})$ has been employed in the third equality.

Next, introduce the quantities

$$\begin{aligned} \mathbf{w}^j:=\sum _{k=1}^{j}A_{n-k}^{(n)}(\mathbf{z}^{k}-\mathbf{z}^{k-1}) \quad \text{ and }\quad B_j:=\frac{1}{A_{n-j}^{(n)}}\quad \text{ for }~1\le j\le n. \end{aligned}$$

It holds that $\mathbf{z}^{j}-\mathbf{z}^{j-1}=B_j(\mathbf{w}^j-\mathbf{w}^{j-1})$ for $2\le j\le n$, and $B_1\ge B_2\ge \cdots \ge B_n$ (according to the monotone property in A1). Then

$$\begin{aligned} J_n&=B_1(\mathbf{w}^1)^T\mathbf{Q}^{(n)}{} \mathbf{w}^1+\sum _{j=2}^nB_j(\mathbf{w}^j-\mathbf{w}^{j-1})^T\mathbf{Q}^{(n)}(\mathbf{w}^j-\mathbf{w}^{j-1})\\&\quad +2\sum _{j=2}^nB_j(\mathbf{w}^j-\mathbf{w}^{j-1})^T\mathbf{Q}^{(n)}{} \mathbf{w}^{j-1}\\&=B_1(\mathbf{w}^1)^T\mathbf{Q}^{(n)}{} \mathbf{w}^1+\sum _{j=2}^nB_j\left[ (\mathbf{w}^j)^T\mathbf{Q}^{(n)}{} \mathbf{w}^j-(\mathbf{w}^{j-1})^T\mathbf{Q}^{(n)}{} \mathbf{w}^{j-1} \right] \\&=B_n(\mathbf{w}^n)^T\mathbf{Q}^{(n)}{} \mathbf{w}^n+\sum _{j=1}^{n-1}(B_j-B_{j+1})(\mathbf{w}^j)^T\mathbf{Q}^{(n)}{} \mathbf{w}^j\\&\ge B_n(\mathbf{w}^n)^T\mathbf{Q}^{(n)}{} \mathbf{w}^n, \end{aligned}$$

because $\mathbf{Q}^{(n)}$ is a positive definite matrix. Hence, the inequality (7.1) is valid since $\mathbf{w}^n=({{\mathcal {D}}}_\tau ^\beta \mathbf{z})^{n-\theta }$ and $B_n=1/A_0^{(n)}$. Similarly, it is easy to trace the remaining parts of [14, Lemma A.1] to check inequality (7.2).

According to [14, Lemma 4.1] and [15, Corollary 2.3], with the maximum time-step ratio $\rho =7/4$, we have

$$\begin{aligned} \frac{1-\theta }{A_0^{(n)}}-\frac{\theta }{A_0^{(n)}-A_1^{(n)}}\ge 0, \end{aligned}$$

which further leads to (2.1) by a simple combination of (7.1) and (7.2).

1.3 Truncation Error Analysis

The truncation errors in (3.19) and (4.16) are given by

$$\begin{aligned}&{{\mathcal {T}}}_u(\mathbf{x}_h,t_{n-\theta }):={{\mathcal {D}}}_t^\alpha u(\mathbf{x}_h,t_{n-\theta })-\left( {{\mathcal {D}}}_\tau ^\alpha u(\mathbf{x}_h,\cdot )\right) ^{n-\theta },\\&{{\mathcal {T}}}_A(\mathbf{x}_h,t_{n-\theta }):={{\mathcal {A}}}_h^{n-\theta }\left\{ u(\mathbf{x}_h,t_{n-\theta })-\left[ (1-\theta )u(\mathbf{x}_h,t_n)+\theta u(\mathbf{x}_h,t_{n-1}) \right] \right\} ,\\&{{\mathcal {T}}}_{{\tilde{u}}}(\mathbf{x}_h,t_{n-\theta }):={{\mathcal {D}}}_t^\beta {{\tilde{u}}}(\mathbf{x}_h,t_{n-\theta })-\left( {{\mathcal {D}}}_\tau ^\beta {{\tilde{u}}}(\mathbf{x}_h,\cdot )\right) ^{n-\theta },\\&{{\mathcal {T}}}_v(\mathbf{x}_h,t_{n-\theta }):={{\mathcal {D}}}_t^\beta v(\mathbf{x}_h,t_{n-\theta })-\left( {{\mathcal {D}}}_\tau ^\beta v(\mathbf{x}_h,\cdot )\right) ^{n-\theta },\\&{{\mathcal {S}}}(\mathbf{x}_h,t_{n-\theta }):=({{\mathcal {A}}}u)(\mathbf{x}_h,t_{n-\theta })-{{\mathcal {A}}}_h^{n-\theta }u(\mathbf{x}_h,t_{n-\theta }), \end{aligned}$$

for $\mathbf{x}_h\in \Omega _h$ and $1\le n\le N$.

We first study the spatial error ${{\mathcal {S}}}(\mathbf{x}_h,t_{n-\theta })$. By the Taylor expansion (see also [26, eq. (31)]), we can take a continuous function ${\xi }^n(\mathbf{x})$ such that

$$\begin{aligned} {\xi }^n(\mathbf{x}_h)&=\left[ \partial _x(p_1\partial _x u)+\partial _y(p_1\partial _y u)\right] (\mathbf{x}_h,t_{n-\theta })\\&\quad -\left\{ \delta _x[(p_1)_h^{n-\theta }\delta _x]+\delta _y[(p_1)_h^{n-\theta }\delta _y]\right\} u(\mathbf{x}_h,t_{n-\theta }), \end{aligned}$$

where $\mathbf{x}_h\in \Omega _h$ and $1\le n\le N$, and $|{\xi }^n(\mathbf{x}_h)|\le C(h_x^2+h_y^2)$ provided that $\Vert u\Vert _{H^4}\le C$ and $p_k(\mathbf{x},\cdot )\in {{\mathcal {C}}}^3(\Omega )$ for $k=1,2$.

Similarly, there is a continuous function $\eta ^n(\mathbf{x})$ such that

$$\begin{aligned} {\eta }^n(\mathbf{x}_h)=\left( p_3\partial _x u+p_4\partial _y u)\right] (\mathbf{x}_h,t_{n-\theta })-\left[ (p_1)_h^{n-\theta }\delta _{{\hat{x}}}+(p_4)_h^{n-\theta }\delta _y\right] u(\mathbf{x}_h,t_{n-\theta }), \end{aligned}$$

where $\mathbf{x}_h\in \Omega _h$ and $1\le n\le N$, and $|{\eta }^n(\mathbf{x}_h)|\le C(h_x^2+h_y^2)$ provided $\Vert u\Vert _{H^3}\le C$ and $|p_k(\mathbf{x},\cdot )|\le C$ for $k=3,4$.

Hence, based on V2 and the regularity assumption (1.7), we have

$$\begin{aligned} \left| {{\mathcal {S}}}(\mathbf{x}_h,t_{n-\theta })\right| ={{\mathcal {O}}}(h_x^2+h_y^2). \end{aligned}$$

(7.3)

By the Taylor expansion with integral remainder, we further get that

$$\begin{aligned} \delta _x\xi ^n(x_{i+\frac{1}{2}},y_j)=\frac{1}{2}\int _0^1\left[ \xi _x^n\left( x_{i+\frac{1}{2}}+\frac{h_x}{2}s,y_j\right) +\xi _x^n\left( x_{i+\frac{1}{2}}-\frac{h_x}{2}s,y_j\right) \right] (1-s)\,\mathrm {d}s, \end{aligned}$$

for $0\le i\le M_x,~1\le j\le M_y-1$, and

$$\begin{aligned} \delta _y\xi ^n(x_i,y_{j+\frac{1}{2}})=\frac{1}{2}\int _0^1\left[ \xi _y^n\left( x_i,y_{j+\frac{1}{2}}+\frac{h_y}{2}s\right) +\xi _y^n\left( x_i,y_{j+\frac{1}{2}}-\frac{h_y}{2}s\right) \right] (1-s)\,\mathrm {d}s, \end{aligned}$$

for $1\le i\le M_x-1,~0\le j\le M_y$. Similar formulations work for $\delta _x\eta ^n(x_{i+\frac{1}{2}},y_j)$ and $\delta _y\eta ^n(x_i,y_{j+\frac{1}{2}})$. Thus, under the assumptions in V2 and (1.7), it is easy to know that

$$\begin{aligned} \Vert \nabla _h {{\mathcal {S}}}(\mathbf{x}_h,t_{n-\theta })\Vert \le C(h_x^2+h_y^2),\quad \mathbf{x}_h\in \Omega _h,~1\le n\le N. \end{aligned}$$

(7.4)

For the temporal truncation errors, according to [20, Lemma 6.1], we have

$$\begin{aligned} \sum _{j=1}^nP_{n-j}^{(n)}\Vert ({{\mathcal {T}}}_A)^{n-\theta }\Vert \le C\tau ^{\min \{2,\gamma \sigma \}}. \end{aligned}$$

(7.5)

Referring to [20, eqs. (7.5), (7.6) and (7.8)], similar to the estimation of $\Vert \nabla _h {{\mathcal {S}}}(\mathbf{x}_h,t_{n-\theta })\Vert$, we have

$$\begin{aligned}&\sum _{j=1}^nP_{n-j}^{(n)}\Vert \nabla _h({{\mathcal {T}}}_A)^{n-\theta }\Vert \le C\tau ^{\min \{2,\gamma \sigma \}}, \end{aligned}$$

(7.6)

$$\begin{aligned}&\sum _{j=1}^nP_{n-j}^{(n)}\Vert \nabla _h({{\mathcal {T}}}_u)^{n-\theta }\Vert \le C\tau ^{\min \{3-\beta ,\gamma \sigma _1\}}, \end{aligned}$$

(7.7)

$$\begin{aligned}&\sum _{j=1}^nP_{n-j}^{(n)}\Vert \nabla _h({{\mathcal {T}}}_{{\tilde{u}}})^{n-\theta }\Vert \le C\tau ^{\min \{3-\beta ,\gamma \sigma _2\}}, \end{aligned}$$

(7.8)

$$\begin{aligned}&\sum _{j=1}^nP_{n-j}^{(n)}\Vert ({{\mathcal {T}}}_{v1})^{n-\theta }\Vert \le C\tau ^{\min \{3-\beta ,\gamma \sigma _3\}}, \end{aligned}$$

(7.9)

$$\begin{aligned}&\sum _{j=1}^nP_{n-j}^{(n)}\Vert \nabla _h({{\mathcal {T}}}_{v2})^{n-\theta }\Vert \le C\tau ^{\min \{2,\gamma \sigma _3\}}, \end{aligned}$$

(7.10)

for $1\le n\le N$, provided that assumptions in V2 and (1.8)–(1.9) are valid.