First we introduce some usual notations and a lemma due to Lopez-Marcos [28] that play a crucial role in convergence analysis of the scheme.

Let \(\Omega_{h}=\{x_{j}\vert0 \leq i \leq N\}\) and \(\Omega_{\tau}=\{ t_{n}\vert0 \leq n \leq M\}\) be uniform partitions of the intervals \([a,b]\) and \([0,L]\), respectively, where \(x_{i}=ih\) and \(t_{n}=n \tau\) with \(\tau=\frac {T}{M}\). Let \(u_{j}^{n}\) be approximation to exact solution at the point \((x_{j},t_{n})\) and \(V=\{v_{j}\vert0\leq j\leq M\}\) and \(W=\{w_{j}\vert0\leq j\leq M\}\) be two grid functions defined on \(\Omega_{h} \). Introduce

$$\begin{gathered} \delta^{2} V=v_{i+1}-2v_{i}+v_{i-1}, \qquad (V,W)=\sum_{i=1}^{M} h v_{i} w_{i},\\ \Vert V \Vert ^{2}=(V,V), \qquad(V_{xx},V)=-(V_{x},V_{x}). \end{gathered} $$

From [28], we have the following important lemma regarding the nonnegative nature of some real quadratic forms possessing a convolution structure.

### Lemma 5.1

*Let*
\(\{w_{n}\}_{n=0}^{\infty}\)
*be a monotonically decreasing sequence of nonnegative real numbers with the property*
\(a_{n+1}+a_{n-1}\geq2 a_{n}\) (\(n\geq1\)), *then for any positive integer*
*K*
*and real vector*
\((V_{1},v_{2},\ldots, V_{K})\in R^{K}\), *we have*

$$\sum_{n=0}^{K-1} \Biggl(\sum _{p=0}^{n}w_{p} V_{n+1-p} \Biggr)V_{n+1} \geq0. $$

Let C be a positive number which assumes different values at different locations and is independent of *i*, *n*, *h* and *τ* such that

$$ \vert u_{tt} \vert \leq C,\qquad \vert u_{xxxx} \vert \leq C\quad \text{for }(x,t)\in\Omega_{h} \times\Omega_{\tau}. $$

(23)

Then, for scheme (7), we have

$$ \begin{gathered}[b] \alpha_{0} \sum _{k=0}^{n} b_{k} \bigl(u(x_{j},t_{n+1-k})-2u(x_{j},t_{n-k})+u(x_{j},t_{n-1-k}) \bigr)+\alpha u\bigl(x_{j},t^{n+1}\bigr) \\ \quad =\frac{\partial^{2} u(x_{j},t^{n+1})}{\partial x^{2}}+ f\bigl(x_{j},t^{n+1}\bigr)+O \bigl(\tau^{2}+\tau h^{2}\bigr) \end{gathered} $$

(24)

and

$$ \alpha_{0} \sum_{k=0}^{n} b_{k} \bigl(u_{j}^{n+1-k}-2u_{j}^{n-k}+u_{j}^{n-1-k} \bigr)+\alpha u_{j}^{n+1}=\frac{\partial^{2} u_{j}^{n+1}}{\partial x^{2}}+ f_{j}^{n+1}, $$

(25)

where \(u(x_{j},t_{n})\) is exact and \(u_{j}^{n}\) is approximate solution at the point \((x_{j},t_{n})\) and \(f_{j}^{n+1}=f(x_{j},t_{n})\).

### Theorem 2

*Let*
\(u(x,t)\)
*and*
\(u_{i}^{n}\)
*be solutions of* (1) *and* (24), *respectively*, *and*
\(u(x,t)\)
*satisfies the smoothness condition* (23), *then for sufficiently small*
*h*
*and*
*τ*, *it holds that*

$$ \bigl\Vert e^{n+1} \bigr\Vert \leq O\bigl( \tau^{2}+\tau h^{2}\bigr), $$

(26)

*where*
\(e_{i}^{n+1}=u(x_{i},t^{n+1})-u_{i}^{n+1}\).

### Proof

To obtain the error equation, we subtract (24) from (25) to get

$$ \alpha_{0} \sum_{k=0}^{n} b_{k} \delta^{2}e_{j}^{n+1-k}+\alpha e_{j}^{n+1}=\bigl(e_{j}^{n+1} \bigr)_{xx}+r_{j}^{n+1}, $$

(27)

where \(r_{j}^{n+1}=O(\tau^{2}+\tau h^{2})\).

Multiplying both sides of (26) by \(he_{j}^{n+1}\) and summing up for j from 1 to *M*, we obtain

$$\begin{aligned} \bigl\Vert e^{n+1} \bigr\Vert ^{2} &= - \frac{\alpha_{0}}{\alpha}\sum_{k=0}^{n} b_{k} \bigl( \delta^{2}e^{n+1-k}, e^{n+1} \bigr)+\frac{1}{\alpha }\bigl(\bigl(e^{n+1}\bigr)_{xx},e^{n+1} \bigr)+\frac{1}{\alpha}\bigl(r^{n+1},e^{n+1}\bigr) \\ &= -\frac{\alpha_{0}}{\alpha}\sum_{k=0}^{n} b_{k} \bigl( \delta^{2}e^{n+1-k}, e^{n+1} \bigr)- \frac{1}{\alpha}\bigl(\bigl(e^{n+1}\bigr)_{x}, \bigl(e^{n+1}\bigr)_{x}\bigr)+\frac {1}{\alpha} \bigl(r^{n+1},e^{n+1}\bigr) \\ &= -\frac{\alpha_{0}}{\alpha}\sum_{k=0}^{n} b_{k} \bigl( \delta^{2}e^{n+1-k}, e^{n+1} \bigr)- \frac{1}{\alpha} \bigl\Vert \bigl(e^{n+1} \bigr)_{x} \bigr\Vert ^{2} +\frac{1}{\alpha} \bigl(r^{n+1},e^{n+1}\bigr). \end{aligned} $$

Rearranging terms, we obtain

$$ \bigl\Vert e^{n+1} \bigr\Vert ^{2}+\frac{\alpha_{0}}{\alpha}\sum _{k=0}^{n} b_{k} \bigl( \delta^{2}e^{n+1-k}, e^{n+1}\bigr)+\frac{1}{\alpha} \bigl\Vert \bigl(e^{n+1}\bigr)_{x} \bigr\Vert ^{2}=\frac{1}{\alpha}\bigl(r^{n+1},e^{n+1}\bigr). $$

Since \(\frac{1}{\alpha} \Vert (e^{n+1})_{x} \Vert ^{2} \geq 0\), therefore

$$ \bigl\Vert e^{n+1} \bigr\Vert ^{2}+\frac{\alpha_{0}}{\alpha}\sum _{k=0}^{n} b_{k} \bigl( \delta^{2}e^{n+1-k}, e^{n+1}\bigr)\leq\frac{1}{\alpha } \bigl(r^{n+1},e^{n+1}\bigr). $$

Then

$$ \begin{gathered} \bigl\Vert e^{n} \bigr\Vert ^{2}+\frac{\alpha_{0}}{\alpha}\sum_{k=0}^{n-1} b_{k} \bigl( \delta^{2}e^{n-k}, e^{n} \bigr)\leq\frac{1}{\alpha}\bigl(r^{n},e^{n}\bigr), \\ \bigl\Vert e^{n-1} \bigr\Vert ^{2}+\frac{\alpha_{0}}{\alpha}\sum _{k=0}^{n-2} b_{k} \bigl( \delta^{2}e^{n-1-k}, e^{n-1}\bigr)\leq\frac{1}{\alpha } \bigl(r^{n-1},e^{n-1}\bigr), \\ \vdots \\ \bigl\Vert e^{2} \bigr\Vert ^{2}+\frac{\alpha_{0}}{\alpha}\sum _{k=0}^{1} b_{k} \bigl( \delta^{2}e^{2-k}, e^{2}\bigr)\leq\frac{1}{\alpha} \bigl(r^{2},e^{2}\bigr), \\ \bigl\Vert e^{1} \bigr\Vert ^{2}+\frac{\alpha_{0}}{\alpha}\sum _{k=0}^{0} b_{k} \bigl( \delta^{2}e^{1-k}, e^{1}\bigr)\leq\frac{1}{\alpha} \bigl(r^{1},e^{1}\bigr). \end{gathered} $$

Adding up all the above inequalities gives

$$ \sum_{k=0}^{n} \bigl\Vert e^{k+1} \bigr\Vert ^{2}+\frac{\alpha _{0}}{\alpha}\sum _{p=0}^{n}\sum_{k=0}^{p} b_{k} \bigl( \delta^{2}e^{p+1-k}, e^{p+1} \bigr)\leq\frac{1}{\alpha}\sum_{k=0}^{n} \bigl(r^{k+1},e^{k+1}\bigr). $$

Using Lemma (5.1), it follows that \(\sum_{p=0}^{n}\sum_{k=0}^{p} b_{k} ( \delta^{2}e^{p+1-k}, e^{p+1})\geq0\) so that we obtain from the last inequality

$$ \sum_{k=0}^{n} \bigl\Vert e^{k+1} \bigr\Vert ^{2} \leq\frac {1}{\alpha}\sum _{k=0}^{n}\bigl(r^{k+1},e^{k+1} \bigr). $$

So

$$\bigl\Vert e^{k+1} \bigr\Vert ^{2} \leq\frac{1}{\alpha} \bigl(r^{k+1},e^{k+1}\bigr). $$

By the Cauchy-Schwarz inequality, we obtain

$$\bigl\Vert e^{k+1} \bigr\Vert ^{2} \leq\frac{1}{\alpha } \bigl(r^{k+1},e^{k+1}\bigr)\leq\frac{1}{\alpha} \bigl\Vert r^{k+1} \bigr\Vert \bigl\Vert e^{k+1} \bigr\Vert . $$

Then

$$\bigl\Vert e^{k+1} \bigr\Vert \leq\frac{1}{\alpha} \bigl\Vert r^{k+1} \bigr\Vert , $$

from where (26) can be very easily deduced. □