## 2 Proposed Numerical Scheme  
The proposed numerical scheme is a two-stage scheme. Both stages of the scheme are implicit. The first stage comprises the information on first- and second-order derivatives of dependent variables in the given differential equation. In the beginning, the procedure of constructing a scheme considers a differential equation  
\[\frac{dy}{dt}=\hat{f}(y(t)) \tag{1}\]  
Subject to the initial condition  
\[y(0)=\alpha \tag{2}\]  
where \(\alpha\) is a constant.  
The first stage of proposing a scheme is to find the solution at an unknown time level. Let it be described as  
\[\overline{y}_{i}=y_{i-1}+hy_{i}^{\prime}+h^{2}y_{i-1}^{\prime\prime} \tag{3}\]  
This second stage (3) can be extended from the Backward Euler method. In this stage (3), \(h\) denotes the step size, which can be expressed as \(h=\frac{L}{N-1}\), where the domain for the solution of (1) is represented as \([0,\ L]\) and \(N\) denotes the number of total time levels.  
Three unknown parameters formed the second stage. Using the Taylor series expansion, the unknown values of three parameters can be determined. The second stage of the proposed scheme is as follows:  
\[y_{i}=y_{i-1}+h\big{\{}a_{1}y_{i}^{\prime}+b_{2}y_{i-1}^{\prime}+a_{3} \overline{y}_{i}^{\prime}\big{\}} \tag{4}\]where \"\(a_{1},b_{2}\)\", and \"\(a_{3}\)\" are three unknown parameters. Expanding \(y_{i-1}\) and \(y_{i-1}^{\prime}\) using the Taylor series, it is obtained:  
\[y_{i}=y_{i}-hy_{i}^{\prime}+\frac{h^{2}}{2}y_{i}^{\prime\prime}-\frac{h^{3}}{6}y _{i}^{\prime\prime}+h\bigg{\{}a_{1}y_{i}^{\prime}+a_{2}y_{i}^{\prime}-a_{2}hy_{ i}^{\prime\prime}+a_{2}\frac{h^{2}}{2}y_{i}^{\prime\prime}+a_{3}\overline{y}_{i}^{ \prime}\bigg{\}} \tag{5}\]  
Substituting Equation (3) into Equation (5) yields  
\[y_{i}=y_{i}-hy_{i}^{\prime}+\frac{h^{2}}{2}y_{i}^{\prime\prime}-\frac{h^{3}}{6} y_{i}^{\prime\prime\prime}+h\bigg{\{}a_{1}y_{i}^{\prime}+a_{2}y_{i}^{\prime}-a_{ 2}hy_{i}^{\prime\prime}+a_{2}\frac{h^{2}}{2}y_{i}^{\prime\prime\prime}+a_{3}y _{i-1}^{\prime}+a_{3}hy_{i}^{\prime\prime}+a_{3}h^{2}y_{i-1}^{\prime\prime \prime}\bigg{\}} \tag{6}\]  
Expanding \(y_{i-1}^{\prime}\) and \(y_{i-1}^{\prime\prime\prime}\) using Taylor series expansion yields  
\[y_{i}=y_{i}-hy_{i}^{\prime}+\frac{h^{2}}{2}y_{i}^{\prime\prime}-\frac{h^{3}}{6 }y_{i}^{\prime\prime\prime}+h\bigg{\{}a_{1}y_{i}^{\prime}+a_{2}y_{i}^{\prime} -a_{2}hy_{i}^{\prime\prime}+a_{2}\frac{h^{2}}{2}y_{i}^{\prime\prime\prime}+a_ {3}y_{i}^{\prime}+a_{3}\frac{h^{2}}{2}y_{i}^{\prime\prime\prime}+a_{3}h^{2}y_{ i}^{\prime\prime\prime}\bigg{\}} \tag{7}\]  
Comparing coefficients of \(hy_{i}^{\prime}\), \(h^{2}y_{i}^{\prime\prime}\), and \(h^{3}y_{i}^{\prime\prime\prime}\) on both sides of Equation (7), it is obtained  
\[1=a_{1}+a_{2}+a_{3} \tag{8}\]  
\[-\frac{1}{2}=-a_{2} \tag{9}\]  
\[\frac{1}{6}=\frac{a_{2}}{2}+\frac{3}{2}a_{3} \tag{10}\]  
By solving Equations (8)-(10), the values of the unknown parameters are obtained as  
\[a_{1}=\frac{5}{9},a_{2}=\frac{1}{2},\ a_{3}=\frac{1}{18} \tag{11}\]  
Thus, the first and second stages of the proposed scheme for discretizing Equation (1) can be expressed as  
\[\overline{y}_{i}=y_{i-1}+h\hat{f}_{i}+h^{2}\hat{f}_{i-1}^{\prime\prime} \tag{12}\]  
\[y_{i}=y_{i-1}+h\bigg{\{}\frac{5}{9}\hat{f}_{i}+\frac{1}{2}\hat{f}_{i-1}-\frac{ 1}{18}\hat{f}_{i}\bigg{\}} \tag{13}\]  
where \(\hat{f}_{i}=\hat{f}(y_{i})\), \(\hat{f}_{i-1}^{\prime}=\frac{df}{dy}\frac{dy}{dt}\bigg{|}_{i-1}\) and \(\hat{\overline{f}}_{i}=\hat{f}(\overline{y}_{i})\).