# New Step Size Control Algorithm for Semi-Implicit Composition ODE Solvers
## 3 Results
### Problem 1: Rossler System
A semi-implicit CD algorithm is as follows:  
\[w^{n+0.5}=w^{n}+\frac{h}{2}\left(-\frac{y^{n}}{(\sqrt{x^{n}x^{n}+y^ {n}y^{n}})^{3}}\right),\] \[z^{n+0.5}=z^{n}+\frac{h}{2}\left(-\frac{x^{n}}{(\sqrt{x^{n}x^{n}+ y^{n}y^{n}})^{3}}\right),\] \[y^{n+0.5}=y^{n}+\frac{h}{2}\left(w^{n+0.5}\right),\] \[x^{n+0.5}=x^{n}+\frac{h}{2}\left(z^{n+0.5}\right),\] \[x^{n+1}=x^{n+0.5}+h\left(z^{n+0.5}\right),\] \[y^{n+1}=y^{n+0.5}+h\left(w^{n+0.5}\right),\] \[z^{n+1}=z^{n+0.5}+h\left(-\frac{x^{n+1}}{\left(\sqrt{x^{n+1}x^{n +1}+y^{n+1}y^{n+1}}\right)^{3}}\right),\] \[w^{n+1}=w^{n+0.5}+h\left(-\frac{y^{n+1}}{\left(\sqrt{x^{n+1}x^{n +1}+y^{n+1}y^{n+1}}\right)^{3}}\right).\]  
Figure 8 shows the performance plots for solving the two-body problem through different adaptive step size composition solvers. Parameter values used in the experiments are presented in Table 6. Dependence between the total number of steps and tolerance can be observed in \[w^{n+0.5}=w^{n}+\frac{h}{2}\left(-\frac{y^{n}}{(\sqrt{x^{n}x^{n}+y^ {n}y^{n}})^{3}}\right),\]0  
\[w^{n+0.5}=w^{n}+\frac{h}{2}\left(-\frac{y^{n}}{(\sqrt{x^{n}x^{n}+y^ {n}y^{n}})^{3}}\right),\]1 Performance plots for two-body problem simulation by solvers of orders 4 (a), 6 (b), and 8 (c). While solving the system with a periodic solution, the proposed ECDM step size control algorithm shows higher performance for low accuracy order composition schemes, while its overall CPU time vs. error features remains the most competitive among all investigated step size control algorithms. The step size plot given in Figure 9 correlates with the periodic behavior of the state variables. The ECDM selects the largest step sizes, which indirectly indicates that the other local error estimation techniques tend to overestimate the error and select fewer step sizes, which makes them less efficient. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multirow{2}{*}{**Initial**} & \multirow{2}{*}{**T**} & \multicolumn{3}{c}{**Step Size**} & \multirow{2}{*}{**Tolerance**} \\ \cline{4-5}  & & & **Min** & **Start** & **Max** \\ \hline
4 & (1; 0; 0; 1) & 50 & \(1\times 10^{-4}\) & \(5\times 10^{-3}\) & 1 & \(1\times 10^{-4}\)-\(1\times 10^{-9}\) \\
6 & (1; 0; 0; 1) & 50 & \(1\times 10^{-4}\) & \(5\times 10^{-3}\) & 1 & \(1\times 10^{-6}\)-\(1\times 10^{-11}\) \\
8 & (1; 0; 0; 1) & 50 & \(1\times 10^{-4}\) & \(5\times 10^{-3}\) & 1 & \(1\times 10^{-5}\)-\(1\times 10^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameter values used in the simulation of two-body problem. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multirow{2}{*}{**Tolerance**} & \multicolumn{4}{c}{**Total Number of Steps**} \\ \cline{3-6}  & & **OCDM** & **DCOM** & **BEE** & **ECDM** \\ \hline \multirow{4}{*}{4} & \(1\times 10^{-4}\) & 177 & 986 & 335 & 156 \\  & \(1\times 10^{-5}\) & 279 & 2121 & 593 & 246 \\  & \(1\times 10^{-6}\) & 441 & 4567 & 1053 & 389 \\  & \(1\times 10^{-7}\) & 697 & 9837 & 1871 & 615 \\  & \(1\times 10^{-8}\) & 1102 & 21,192 & 3324 & 972 \\  & \(1\times 10^{-9}\) & 1744 & 45,655 & 5910 & 1539 \\ \hline \multirow{4}{*}{6} & \(1\times 10^{-6}\) & 309 & 442 & 745 & 289 \\  & \(1\times 10^{-7}\) & 427 & 699 & 1179 & 401 \\  & \(1\times 10^{-8}\) & 593 & 1106 & 1868 & 547 \\  & \(1\times 10^{-9}\) & 823 & 1751 & 2959 & 759 \\  & \(1\times 10^{-10}\) & 1143 & 2774 & 4689 & 1048 \\  & \(1\times 10^{-11}\) & 1587 & 4395 & 7430 & 1452 \\ \hline \multirow{4}{*}{8} & \(1\times 10^{-5}\) & 88 & 234 & 197 & 83 \\  & \(1\times 10^{-6}\) & 113 & 322 & 288 & 106 \\ \cline{1-1}  & \(1\times 10^{-7}\) & 146 & 444 & 421 & 136 \\ \cline{1-1}  & \(1\times 10^{-8}\) & 187 & 611 & 616 & 175 \\ \cline{1-1}  & \(1\times 10^{-9}\) & 241 & 849 & 904 & 226 \\ \cline{1-1}  & \(1\times 10^{-10}\) & 310 & 1179 & 1325 & 290 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Dependence between the total number of steps and tolerance while solving two-body problem. Figure 9: Graphs of step size behavior for different controllers under investigation for two-body problem.