## 4 Truncation Error Estimation
According to (9), the local truncation error can be written as  
\[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]  
By using Taylor series, the terms \(y_{i-1},y_{i-1/2},y_{i+1},y_{i+1/2}\) and \(M_{i-1/2}^{\prime},M_{i+1/2}^{\prime}\) can be expanded around the point \(x_{i}\), hence we have  
\[t_{i}^{1}=\sum_{j=0}^{6}k_{j}^{1}y_{i}^{(j)}+O(h^{7}),\]  
where  
\[k_{0}^{1}=0;k_{1}^{1}=h(\lambda_{i-1}A_{i}-\lambda_{i}A_{i-1})+ \frac{h}{2}(\lambda_{i-1}A_{i}B_{i-1}-\lambda_{i}A_{i-1}B_{i});\] \[k_{2}^{1}=-\frac{3h^{2}}{4}(\lambda_{i-1}A_{i}+\lambda_{i}A_{i-1 })-\frac{h^{2}}{8}(\lambda_{i-1}A_{i}B_{i-1}+\lambda_{i}A_{i-1}B_{i})+\frac{A_ {i}(2-B_{i-1})}{\lambda_{i-1}}+\frac{A_{i-1}(2-B_{i})}{\lambda_{i}};\] \[k_{3}^{1}=\frac{7h^{3}}{24}(\lambda_{i-1}A_{i}-\lambda_{i}A_{i-1 })+\frac{h^{3}}{48}(\lambda_{i-1}A_{i}B_{i-1}-\lambda_{i}A_{i-1}B_{i})-\frac{ hA_{i}(2-B_{i-1})}{2\lambda_{i-1}}+\frac{hA_{i-1}(2-B_{i})}{2\lambda_{i}};\] \[k_{4}^{1}=-\frac{15h^{4}}{192}(\lambda_{i-1}A_{i}+\lambda_{i}A_{i -1})-\frac{h^{4}}{384}(\lambda_{i-1}A_{i}B_{i-1}+\lambda_{i}A_{i-1}B_{i})+ \frac{h^{2}A_{i}(2-B_{i-1})}{8\lambda_{i-1}}+\frac{h^{2}A_{i-1}(2-B_{i})}{8 \lambda_{i}};\] \[k_{5}^{1}=\frac{31h^{5}}{1920}(\lambda_{i-1}A_{i}-\lambda_{i}A_{i -1})+\frac{h^{5}}{3840}(\lambda_{i-1}A_{i}B_{i-1}-\lambda_{i}A_{i-1}B_{i})- \frac{h^{3}A_{i}(2-B_{i-1})}{48\lambda_{i-1}}+\frac{h^{3}A_{i-1}(2-B_{i})}{48 \lambda_{i}};\]  
\[k_{6}^{1}=-\frac{63h^{6}}{23040}(\lambda_{i-1}A_{i}+\lambda_{i}A_{i-1})-\frac {h^{6}}{46080}(\lambda_{i-1}A_{i}B_{i-1}+\lambda_{i}A_{i-1}B_{i})+\frac{h^{4} A_{i}(2-B_{i-1})}{384\lambda_{i-1}}+\frac{h^{4}A_{i-1}(2-B_{i})}{384\lambda_{i}}.\]  
Note that \(A_{i}=2\sinh(\lambda_{i}h),B_{i}=2\cosh(\lambda_{i}h)\), since the hyperbolic functions can be expanded in  
\[\sinh(x)=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}=x+\frac{x^{3}}{3!}+\frac{ x^{5}}{5!}+\cdots;\cosh(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}=1+\frac{x^{2}}{2!} +\frac{x^{4}}{4!}+\cdots.\]  
By expanding the terms \(A_{i-1},A_{i},B_{i-1}\) and \(B_{i}\) in series, and by a tedious but straightforward calculation, \(k_{j}^{1}(j=1,2,3,4,5,6)\) can be written as  
\[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]0 \[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]1  
Similarly, the local truncation error of (10) can be written as  
\[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]2where  
\[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]3 \[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]4 \[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]5 \[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]6 \[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]7  
\[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]8 \[t_{i}^{1}=\alpha_{i}^{1}\tilde{y}_{i-1}+\beta_{i}^{1}\tilde{y}_{i}+\gamma_{i}^{1 }\tilde{y}_{i+1}-\delta_{i}^{1}.\]9 \[t_{i}^{1}=\sum_{j=0}^{6}k_{j}^{1}y_{i}^{(j)}+O(h^{7}),\]0 \[t_{i}^{1}=\sum_{j=0}^{6}k_{j}^{1}y_{i}^{(j)}+O(h^{7}),\]1 \[t_{i}^{1}=\sum_{j=0}^{6}k_{j}^{1}y_{i}^{(j)}+O(h^{7}),\]2 \[t_{i}^{1}=\sum_{j=0}^{6}k_{j}^{1}y_{i}^{(j)}+O(h^{7}),\]3  
Since the trigonometric functions can be expressed in  
\[t_{i}^{1}=\sum_{j=0}^{6}k_{j}^{1}y_{i}^{(j)}+O(h^{7}),\]4  
By expanding the trigonometric functions of \(k_{j}^{2}(j=1,2,\cdots,6)\) in series, then \(k_{j}^{2}(j=1,2,\cdots,6)\) can be written as  
\[t_{i}^{1}=\sum_{j=0}^{6}k_{j}^{1}y_{i}^{(j)}+O(h^{7}),\]5 \[t_{i}^{1}=\sum_{j=0}^{6}k_{j}^{1}y_{i}^{(j)}+O(h^{7}),\]6 \[t_{i}^{1}=\sum_{j=0}^{6}k_{j}^{1}y_{i}^{(j)}+O(h^{7}),\]7  
Then the local truncation error of (10) is \(O(h^{4})\).