In this note, we revisit the two-step backward differentiation formula (BDF2) with variable time-steps for solving the following reaction-diffusion equation:

$ \begin{equation} \begin{aligned} &\partial_t u = \Delta u +\kappa u +f(x, t), &&x\in\Omega, t\in(0, T], \\ &u(x, 0) = u_0(x), && x\in\bar{\Omega}, \\ &u(x, t) = 0, &&x\in\partial\Omega, t\in[0, T], \end{aligned} \end{equation} $

(1)

where the reaction coefficient $ \kappa \in \mathbb{R} $, and $ \Omega $ is a bounded domain.

Set the generally nonuniform time levels $ 0 = t_0<t_1<t_2<\cdots<t_N = T $ with the $ k $th time-step size $ \tau_k : = t_k-t_{k-1} $, the maximum step size $ \tau : = \max_{1\le k\le N}\tau_k $, and the adjacent time-step ratio

$ r_k = \frac{\tau_k}{\tau_{k-1}}, \quad \quad \quad \quad 2\leq k \leq N. $

The BDF2 scheme with variable time-step is given as

$ \begin{align} \mathcal{D}_2 u^n = \Delta u^n +\kappa u^n +f^n, \quad \text{for} \quad 1\leq n\leq N, \end{align} $

(2)

where the BDF2 formula can be unified to the following discrete convolution form

$ \begin{equation} \mathcal{D}_2 u^n : = \sum\limits_{k = 1}^n b_{n-k}^{(n)} \nabla_\tau u^k, \quad \quad n\geq 1, \end{equation} $

(3)

which is simulated by the discrete fractional derivatives [1-4]. Here the discrete convolution kernels $ b_{n-k}^{(n)} $ are given as $ b_{0}^{(1)} : = 1/\tau_1 $ and for $ n>1 $,

$ \begin{align} b_0^{(n)} = \frac{1+2r_n}{\tau_n(1+r_n)}, \quad b_1^{(n)} = - \frac{r_n^2}{\tau_n(1+r_n)} \quad \text{and} \quad b_j^{(n)} = 0 \quad \quad\text{ for } 2\leq j \leq n-1. \end{align} $

(4)

For the problem (1) with $ \kappa = 0 $, twenty years ago Becker [5] (one also refers to Thomée's classical book [6, Lemma 10.6]) presented that if the adjacent step ratios $ r_k $ satisfy $ 0< r_k \leq (2+\sqrt{13})/3\approx 1.868 $, the discrete solution can be bounded by

$ \left\|{u^n}\right\| \leq C \exp(C\Gamma_n)\big(\left\|{u_0}\right\| + \sum\limits_{j = 1}^n \tau_j \left\|{f^j}\right\| \big)\quad \text{for} \quad n\geq 1, $

where $ C $ is a positive constant and $ \Gamma_n : = \sum_{k = 2}^{n-2} \max\{0, r_k-r_{k+2}\} $. As pointed out in [6] and [7], the magnitudes of $ \Gamma_n $ can be zero, bounded [6, pp. 175] and unbounded [7, Remark 4.1] by selecting certain step-ratio sequence and vanishing step sizes. After that, Emmrich [8] improves the Becker's constrained condition to $ 0\leq r_k \leq 1.91 $, but still remains the undesirable factor $ \exp(C\Gamma_n) $ in the $ L_2 $-norm stability. Chen et al. circumvent the factor $ \exp(C\Gamma_n) $ in Becker's estimate with a bounded factor $ \exp(Ct_n) $ with $ 0\leq r_k \leq 1.53 $, but lack the estimate in the ideal case of $ \Gamma_n = 0 $. Recently, by using the technique of the discrete orthogonal convolution (DOC) kernels, a nice and interesting work [9] obtains the convergence

$ \begin{align} &\left\|{u(t_n)-u^n}\right\|\\ \leq& 2\exp\big(4\kappa t_{n-1}\big) \big(\left\|{u_0\!-\!u^0}\right\| +2t_n\int^{t_1}_{0} \|\partial_{tt} u\| \, \mathrm{d} t + 3 t_n \max\limits_{1\leq k\leq n}\tau_k \int_{t_{k-1}}^{t_k} \|\partial_{ttt} u\| \, \mathrm{d} t\big) \end{align} $

(5)

with $ 0\leq r_k \leq 3.561 $. Here the DOC kernels are defined by

$ \begin{align} & \sum\limits_{j = k}^n \theta_{n-j}^{(n)} b_{j-k}^{(j)} = \delta_{nk}, \quad \text{for all } 1\leq k \leq n, \end{align} $

(6)

where $ \delta_{nk} $ represents the Kronecker delta symbol with $ \delta_{nk} = 1 $ if $ n = k $ and $ \delta_{nk} = 0 $ if $ n\neq k $. One can see that the right-hand-side second term is the first-order convergence when $ t_n $ is large. If the second-order convergence is obtained, it suffers from a restriction condition $ |\mathfrak{R}_p| \leq N_0 \ll N $ with the index set defined by

$ \begin{equation} \mathfrak{R}_p = \left \{ k \;\left | \;1+\sqrt{2} \leq r_k \leq (3+\sqrt{17})/2 \right.\right\}. \end{equation} $

(7)

In this note, by introducing the novel conception of the discrete complementary convolution (DCC) kernels, we achieve the sharp second-order convergence for BDF2 scheme and update the adjacent time-step restriction condition to

$ {\bf{A1}}: 0< r_k \leq r_{\max} = \frac{1}{6} \left(\sqrt[3]{1196\!-\!12\sqrt{177}}+\sqrt[3]{1196\!+\!12\sqrt{177}}\right)+\frac43 \approx 4.864{5}, {\rm{for}}\\ 2\leq k \leq N, $

which does not suffer from the restriction condition $ |\mathfrak{R}_p| \leq N_0 \ll N $. Here $ r_{\max} $ is the positive real solution of $ x^3 = (2x+1)^2 $.

To obtain stability analysis of the discrete scheme (2), it is natural to find a DCC kernel $ p_{n-j}^{(n)} $ such that

$ \begin{equation} \sum\limits_{j = 1}^np_{n-j}^{(n)}\mathcal{D}_2 u^j = \sum\limits_{j = 1}^np_{n-j}^{(n)} \sum\limits_{l = 1}^j b_{j-l}^{(j)} \nabla_\tau u^l = \sum\limits_{l = 1}^n\nabla_\tau u^l \sum\limits_{j = l}^np_{n-j}^{(n)}b_{j-l}^{(j)} = u^n - u^0, \quad \forall n\geq 1. \end{equation} $

(8)

One can see that if the identity (8) holds for all $ n\geq 1 $, it only requires

$ \begin{equation} \sum\limits_{j = k}^n p_{n-j}^{(n)} b_{j-k}^{(j)} \equiv1, \quad \forall 1\leq k \leq n, \; 1 \leq n \leq N. \end{equation} $

(9)

The first main contribution in our paper is establishing the positive semi-definiteness of BDF2 convolution kernels, which produces the constrain condition A1 on the adjacent time-step ratios.

Lemma 0.1  Assume the time step ratios $ r_k $ satisfy A1. For any real sequence $ \{w_k\}_{k = 1}^n $, it holds that

$ \begin{align} &2w_k \sum\limits_{j = 1}^k b^{(k)}_{k-j} w_j \geq \frac{r_{k+1}}{(1+r_{k+1})}\frac{w_k^2}{\epsilon_*\tau_k} - \frac{r_k}{ (1+r_k)}\frac{w_{k-1}^2 }{\epsilon_*\tau_{k-1}}, \quad k \geq 2, \end{align} $

(10)

$ \begin{align} &2\sum\limits_{k = 1}^nw_k \sum\limits_{j = 1}^k b^{(k)}_{k-j} w_j \geq 0, \quad \text{for } n \geq 1, \end{align} $

(11)

where the positive constant $ \epsilon_* = \frac16\sqrt[3]{12} \big(\sqrt[3]{\sqrt{177} + 9} - \sqrt[3]{\sqrt{177} - 9} \;\big) \approx 0.4534 $ is taken.

The positive semi-definitiveness of $ \theta^{(n)}_{n-k} $ is derived by the positive semi-definitiveness of $ b^{(n)}_{n-k} $.

Lemma 0.2  If the BDF2 kernels $ b_{n-k}^{(n)} $ defined in (4) are positive semi-definite, then the DOC kernels $ \theta^{(n)}_{n-k} $ defined in (6) are also positive semi-definite. This is, it holds for any real sequence $ \{\omega_j\}_{j = 1}^n $ that

$ \sum\limits_{k = 1}^n\omega_k\sum\limits_{j = 1}^k\theta_{k-j}^{(k)}\omega_j \geq 0, \quad \forall n\geq 1. $

A immediate product of the semi-positive definiteness of the BDF2 kernels is the following energy stability for BDF2 scheme (2) (one also refers to [9]).

Theorem 0.1  Assume the condition A1 holds and $ \kappa\leq 0 $, then the discrete solution $ u^n $ to the BDF2 scheme (2) with variable time steps satisfies

$ \begin{equation} \partial_\tau E^k\leq 2\langle f^k, \partial_\tau u^k\rangle, \quad \forall k\geq 1. \end{equation} $

(12)

Furthermore, the energy has the following estimate:

$ \begin{equation} \sqrt{E^n}\leq \sqrt{E^0}+ 4C_\Omega( \sum\limits_{k = 1}^{n}\|\nabla_\tau f^{k}\|+\|f^0\|), \quad \forall n\geq 1. \end{equation} $

(13)

Here the (modified) discrete energy $ E^k $ is defined by

$ \begin{equation} E^k: = \frac{r_{k+1}\tau_k}{\epsilon^*(1+r_{k+1})}\|\partial_\tau u^k\|^2+|u^k|_1^2-\kappa \| u^k\|^2, \quad \text{for}\ \kappa\leq 0 \ \text{and} \ k\geq 1, \end{equation} $

(14)

where the initial energy $ E^0: = |u^0|_1^2-\kappa\left\|{ u^0}\right\|^2 $, $ \partial_\tau u^k = \nabla_\tau u^k /\tau_k $ and $ \epsilon_* = \frac16\sqrt[3]{12} \big(\sqrt[3]{\sqrt{177} + 9} - \sqrt[3]{\sqrt{177} - 9} \;\big) \approx 0.4534 $ and $ \left\|{\cdot}\right\| $ represent the inner product and norm in the space $ L^2(\Omega) $.

The second main contribution in our paper is that we achieve the sharp error estimate. To obtain the sharp error estimate, we first establish the stability and consistency results for the BDF2 scheme (2). Before that, several useful properties of DOC and DCC kernels are presented.

Lemma 0.3 [9]  The DOC kernels $ \theta_{n-j}^{(n)} $ have the following properties:

$ \begin{align} \sum\limits_{j = 1}^{n} \theta_{n-j}^{(n)} & = \tau_n, \quad \quad \text{for } n\geq 1, \end{align} $

(15)

$ \begin{align} \sum\limits_{k = 1}^{n} \sum\limits_{j = 1}^{k}\theta_{k-j}^{(k)} & = t_n , \quad \quad \text{for } n\geq 1. \end{align} $

(16)

Lemma 0.4  The DOC kernel $ \theta_{n-j}^{(n)} $ can be explicitly represented by

$ \begin{equation} \begin{aligned} \theta_{n-k}^{(n)} & = \left \{\begin{array}{l} \frac{\tau_n (1+r_k)}{1+2r_k} \prod\limits_{i = k+1}^n \frac{r_i}{1+2r_i}, \quad\quad \text{ for } 2\leq k\leq n. \\ \tau_n \prod\limits_{i = k+1}^n \frac{r_i}{1+2r_i}, \quad\quad \text{ for } k = 1. \end{array} \right. \end{aligned} \end{equation} $

(17)

We remark that the results in Lemma 0.4 is very important for the following convergence analysis and the bound of DCC kernels. Here we emphasize on the connection of DCC and DOC kernels, and present the properties of DCC kernels by propositions as follows.

Proposition 0.1  Let the DCC kernels $ p^{(n)}_{n-j} $ and the DOC kernels $ \theta^{(n)}_{n-j} $ be defined by (9) and (6) respectively. Then, the DCC kernel and DOC kernel hold

$ \begin{align} p^{(n)}_{n-j}& = \sum\limits_{l = j}^n \theta^{(l)}_{l-j}, \quad \forall 1\leq j\leq n, \end{align} $

(18)

$ \begin{align} \theta^{(n)}_{n-j} & = p^{(n)}_{n-j}-p^{(n-1)}_{n-1-j}, \quad \forall 1\leq j\leq n, \end{align} $

(19)

where $ p^{(n)}_{-1}: = 0, \forall n\geq 0 $ is defined.

Proposition 0.2  Let $ \tau $ be the maximum time-step size and the time-step ratios satisfy $ 0<r_k\leq r_{*} $, where $ r_{*} $ is any given positive constant. The DCC kernels $ p^{(n)}_{n-k} $ defined in (9) satisfy

$ \begin{align} &p^{(n)}_{n-j} = \sum\limits_{k = j}^n\frac{\tau_k(1+r_k)}{1+2r_k}\prod\limits_{i = j+1}^k \frac{r_i}{1+2r_i}, \quad 2\leq j\leq n, \end{align} $

(20)

$ \begin{align} &p^{(n)}_{n-1} = \sum\limits_{k = 1}^n\tau_k\prod\limits_{i = 2}^k \frac{r_i}{1+2r_i}, \end{align} $

(21)

$ \begin{align} &\sum\limits_{j = 1}^n p^{(n)}_{n-j} = t_n, \end{align} $

(22)

$ \begin{align} &p^{(n)}_{n-j}\leq \sum\limits_{k = j}^n\tau_k\left(\frac{r_{*}}{1+2r_{*}}\right)^{k-j} \leq \sum\limits_{k = j}^n\frac{\tau_k}{2^{k-j}}\leq 2\tau, \end{align} $

(23)

where $ \prod_{i = j+1}^k = 1 $ for $ j\geq k $ is defined.

To obtain the stability of the BDF2 scheme (2), we introduce a discrete Grönwall inequality for the following $ L^2 $-norm estimate.

Lemma 0.5  Assume $ \lambda>0 $ and the sequences $ \{v_j\}_{j = 1}^N $ and $ {\{\eta_j\}_{j = 0}^N} $ are nonnegative. If

$ v_n \leq \lambda \sum\limits_{j = 1}^{n-1} \tau_j v_j + \sum\limits_{j = 0}^n \eta_j, \quad \text{for} \quad 1\leq n \leq N, $

then it holds

$ v_n \leq \exp\big(\lambda t_{n-1}\big) \sum\limits_{j = 0}^n \eta^j, \quad \text{for} \quad 1\leq n \leq N. $

We now present the stability result of the BDF2 scheme (2).

Theorem 0.2  If the BDF2 kernels $ b_{n-j}^{(n)} $ defined in (4) are positive semi-definite (or condition A1 holds), the discrete solution $ u^n $ of the BDF2 scheme (2) is unconditionally stable in the $ L^2 $-norm. If $ \kappa>0 $ and the maximum time-step size $ \tau \leq 1/(4\kappa) $, it holds

$ \begin{align} &{\left\|{u^n}\right\| \leq 2\exp\big(4\kappa t_{n-1}\big) \left(\left\|{u^0}\right\| + 2\sum\limits_{j = 1}^{n} p_{n-j}^{(n)} \left\|{f^j}\right\|\right), \quad \text{for } 1\leq n\leq N.} \end{align} $

(24)

If $ \kappa\leq 0 $, it holds

$ \begin{align} &\left\|{u^n}\right\|\leq \left\|{u^0}\right\| + 2\sum\limits_{j = 1}^{n}p_{n-j}^{(n)} \left\|{f^j}\right\|, \quad \text{for } 1\leq n\leq N. \end{align} $

(25)

Lemma 0.6  Set

$ \begin{equation} \begin{aligned} G^l &: = -\frac12\int_{t_{l-1}}^{t_l} (t-t_{l-1})^2\partial_{ttt} u \, \mathrm{d} t, 1\leq l \leq N, \\ R^j &: = -\frac1{2}b^{(j)}_1\tau_{j-1}\int_{t_{j-1}}^{t_j}(2(t-t_{j-1})+\tau_{j-1})\partial_{ttt} u \, \mathrm{d} t, \quad 2 \leq j\leq {N}, \\ R^1&: = \frac1{2\tau_1}\int_0^{t_1} t^2\partial_{ttt}u \, \mathrm{d} t{-\frac1{\tau_1}\int_0^{t_1} su_{tt}(s) \, \mathrm{d} s}. \end{aligned} \end{equation} $

(26)

The truncation error $ \eta^j: = \mathcal{D}_2 u(t_j)-\partial_t u(t_j) \;(1\leq j\leq N) $ can be expressed the following form

$ \begin{equation} \eta^j = \sum\limits_{l = 1}^{j}b^{(j)}_{j-l}G^l+R^j, \quad 1\leq j\leq {N}. \end{equation} $

(27)

Theorem 0.3  Assume the conditions in Theorem 0.2 hold, and the truncated error can be expressed by

$ \begin{equation} f^n : = \sum\limits_{k = 1}^n b_{n-k}^{(n)} G^k + R^n, \end{equation} $

(28)

where $ G^k $ and $ R^n $ are given in (26). If $ \kappa>0 $ and the maximum time-step size $ \tau \leq 1/(4\kappa) $, it holds

$ \begin{align} \left\|{u^n}\right\| \leq 2\exp\big(4\kappa t_{n-1}\big) \left(\left\|{u^0}\right\| + 2\sum\limits_{k = 1}^{n} \left\|{G^k}\right\|+2\sum\limits_{k = 1}^{n} p_{n-k}^{(n)} \|R^k\|\right), \quad \text{for } 1\leq n\leq N. \end{align} $

(29)

If $ \kappa\leq 0 $, it holds

$ \begin{align} \left\|{u^n}\right\|\leq \left\|{u^0}\right\| + 2\sum\limits_{k = 1}^{n} \left\|{G^k}\right\|+2\sum\limits_{k = 1}^{n} p_{n-k}^{(n)} \|R^k\|, \quad \text{for } 1\leq n\leq N. \end{align} $

(30)

Finally, applying the Lemmas 0.6, Proposition 0.2 and Theorem 0.3, we achieve the sharp error estimate.

Theorem 0.4  Let $ u(t, x) $ be the solution to problem (1). If the BDF2 kernels $ b_{n-k}^{(n)} $ defined in (4) are positive semi-definite (or the condition A1 holds), then the solution $ u^n $ to BDF2 scheme (2) is convergent in the $ L^2 $-norm. If $ \kappa>0 $ and the maximum time-step size $ \tau<1/(4\kappa) $, it holds

$ \begin{align} \left\|{u(t_n)-u^n}\right\| \leq& 2\exp\big(4\kappa t_{n-1}\big) \left(\left\|{u(0)-u^0}\right\| +2(\tau_1+\tau)\int^{t_1}_{0} \|\partial_{tt} u\| \, \mathrm{d} t+ \sum\limits_{k = 1}^{n} \tau_k^2\int_{t_{k-1}}^{k_l}\| \partial_{ttt} u\| \, \mathrm{d} t\right. \\ &\left.+ {2} t_n \max\limits_{{1}\leq k\leq n}\tau_k \int_{t_{k-1}}^{t_k} \|\partial_{ttt} u\| \, \mathrm{d} t\right), \quad \text{for } 1\leq n\leq N. \end{align} $

(31)

If $ \kappa\leq 0 $, it holds

$ \begin{align} \left\|{u(t_n)-u^n}\right\| \leq& \left\|{u(0)-u^0}\right\| +2(\tau_1+\tau)\int^{t_1}_{0} \|\partial_{tt} u\| \, \mathrm{d} t + \sum\limits_{k = 1}^{n} \tau_k^2\int_{t_{k-1}}^{k_l}\| \partial_{ttt} u\| \, \mathrm{d} t \\ &+ {2}t_n \max\limits_{{1}\leq k\leq n}\tau_k \int_{t_{k-1}}^{t_k} \|\partial_{ttt} u\| \, \mathrm{d} t, \quad \text{for } 1\leq n\leq N. \end{align} $

(32)

More details can be found in [10].