## 3 Order Conditions of the DIRKT Method  
The algebraic order conditions for the RKT formula up to order seven given in [PERSON] et al. [17] as follows:  
**The order terms for y:**  
4 th-order  
\[\sum b_{i}=\frac{1}{24}, \tag{3}\]  
5 th-order  
\[\sum b_{i}c_{i}=\frac{1}{120}, \tag{4}\]  
6 th-order  
\[\sum b_{i}c_{i}^{2}=\frac{1}{360}, \tag{5}\]  
7 th-order  
\[\sum b_{i}c_{i}^{3}=\frac{1}{840}, \tag{6}\]  
**The order terms for y':**  
3 rd-order  
\[\sum b_{i}^{\prime}=\frac{1}{6}, \tag{7}\]  
4 th-order  
\[\sum b_{i}^{\prime}c_{i}=\frac{1}{24}, \tag{8}\]  
5 th-order  
\[\sum b_{i}^{\prime}c_{i}^{2}=\frac{1}{60}, \tag{9}\]  
6 th-order  
\[\sum b_{i}^{\prime}c_{i}^{3}=\frac{1}{120}, \tag{10}\]  
7 th-order  
\[\sum b_{i}^{\prime}c_{i}^{4}=\frac{1}{210},\sum b_{i}^{\prime}a_{ij}=\frac{1 }{5040} \tag{11}\]  
**The order terms for y'':**  
2 nd-order  
\[\sum b_{i}^{\prime\prime}=\frac{1}{2}, \tag{12}\]  
3 rd-order  
\[\sum b_{i}^{\prime\prime}c_{i}=\frac{1}{6}, \tag{13}\]  
4 th-order  
\[\sum b_{i}^{\prime\prime}c_{i}^{2}=\frac{1}{12}, \tag{14}\]  
5 th-order  
\[\sum b_{i}^{\prime\prime}c_{i}^{3}=\frac{1}{20}, \tag{15}\]  
6 th-order  
\[\sum b_{i}^{\prime\prime}c_{i}^{4}=\frac{1}{30},\quad\sum b_{i}^{\prime\prime }a_{ij}=\frac{1}{720}, \tag{16}\]  
7 th-order  
\[\sum b_{i}^{\prime\prime}c_{i}^{5}=\frac{1}{42},\quad\sum b_{i}^{\prime\prime }a_{ij}\,c_{j}=\frac{1}{5040},\sum b_{i}^{\prime\prime}\,c_{i}a_{ij}=\frac{1 }{1008} \tag{17}\]  
**The order terms for y\({}^{\prime\prime\prime}\):**  
1 st-order  
\[\sum b_{i}^{\prime\prime\prime}=1, \tag{18}\]  
2 nd-order  
\[\sum b_{i}^{\prime\prime\prime}c_{i}=\frac{1}{2}, \tag{19}\]  
3 th-order  
\[\sum b_{i}^{\prime\prime\prime}c_{i}^{2}=\frac{1}{3}, \tag{20}\]  
4 th-order  
\[\sum b_{i}^{\prime\prime\prime}c_{i}^{3}=\frac{1}{4}, \tag{21}\]  
5 th-order  
\[\sum b_{i}^{\prime\prime\prime}c_{i}^{4}=\frac{1}{5},\quad\sum b_{i}^{\prime \prime\prime}a_{ij}=\frac{1}{120}, \tag{22}\]  
6 th-order  
\[\sum b_{i}^{\prime\prime\prime}c_{i}^{5}=\frac{1}{6},\quad\sum b_{i}^{\prime \prime\prime}a_{ij}c_{j}=\frac{1}{720},\sum b_{i}^{\prime\prime\prime}c_{i}a_ {ij}=\frac{1}{144}, \tag{23}\]  
7 th-order  
\[\sum b_{i}^{\prime\prime\prime}c_{i}^{6}=\frac{1}{7},\quad\sum b_{i}^{\prime \prime\prime}c_{i}a_{ij}c_{j}=\frac{1}{840}, \tag{24}\]  
\[\sum b_{i}^{\prime\prime\prime}a_{ij}c_{j}^{2}=\frac{1}{2520},\quad\sum b_{i} ^{\prime\prime\prime}c_{i}^{2}a_{ij}=\frac{1}{168} \tag{25}\]  
**4. Construction of the DIRKT Methods**  
By the order conditions stated in Section 3 above which derived by [PERSON] et al. [17] we proceed to construct diagonally implicit Runge-Kutta type method. The local truncated error for the \(p\) order DIRKT technique is defined as follows:  
\[\parallel\tau_{g}^{(p+1)}\parallel_{2}=\left(\sum_{i=1}^{n_{p}+1}\left(\tau_ {i}^{(p+1)}\right)^{2}+\sum_{i=1}^{n_{p}^{\prime}+1}\left(\tau_{i}^{\prime(p+ 1)}\right)^{2}+\sum_{i=1}^{n_{p}^{\prime\prime}+1}\left(\tau_{i}^{\prime \prime(p+1)}\right)^{2}+\sum_{i=1}^{n_{p}^{\prime\prime\prime}+1}\left(\tau_{ i}^{\prime\prime\prime(p+1)}\right)^{2}\right)^{\frac{1}{2}} \tag{26}\]  
where \(\tau^{(p+1)},\tau^{\prime(p+1)},\tau^{\prime\prime(p+1)}\) and \(\tau^{\prime\prime\prime(p+1)}\) are the local truncation error terms for \(u,u^{\prime},u^{\prime\prime}\) and \(u^{\prime\prime\prime}\) respectively, \(\tau_{g}^{(p+1)}\) is the global local truncation error.