# New Step Size Control Algorithm for Semi-Implicit Composition ODE Solvers
## 3 Results
### Problem 1: Rossler System  
The Rossler system is well-known as being one of the simplest ODE systems with chaotic behavior and a fine test problem for numerical ODE solvers [16]. In this paragraph, all simulations were carried out with parameters \(a=0.2,\ b=0.2,\ c=5.7\), which correspond to a chaotic regime. \[ \begin{array}{l}\dot{x}=-y-z,\\ \dot{y}=x+ay,\\ \dot{z}=b+z(x-c)\end{array} \]  
A semi-implicit CD algorithm (7) for the Rossler problem can be formulated as follows:  
\[ \begin{array}{l}y^{n+0.5}=y^{n}+\frac{h}{2}(x^{n}+ay^{n}),\\ z^{n+0.5}=z^{n}+\frac{h}{2}(b+z^{n}x^{n}-cz^{n}),\\ x^{n+1}=x^{n}+h(-y^{n+0.5}-z^{n+0.5}),\\ z^{n+1}=\frac{z^{n+0.5}+\frac{h}{2}b}{1-z^{n+1}+c\frac{h}{2}},\\ y^{n+1}=\frac{y^{n+0.5}+\frac{h}{2}x^{n+1}}{1-a^{\frac{h}{2}}}.\end{array} \] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multirow{2}{*}{**Initial**} & \multirow{2}{*}{**T**} & \multicolumn{4}{c}{**Step Size**} & \multirow{2}{*}{**Tolerance**} \\ \cline{3-3} \cline{5-6}  & & & **Min** & **Start** & **Max** & \\ \hline
4 & (1.15; 0) & 15 & \(1\times 10^{-5}\) & \(1\times 10^{-4}\) & \(1\) & \(1\times 10^{-5}\)–\(1\times 10^{-10}\) \\
6 & (1.15; 0) & 15 & \(1\times 10^{-5}\) & \(1\times 10^{-4}\) & \(1\) & \(1\times 10^{-5}\)–\(1\times 10^{-10}\) \\
8 & (1.52; 0) & 15 & \(1\times 10^{-5}\) & \(1\times 10^{-4}\) & \(1\) & \(1\times 10^{-4}\)–\(1\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter values used in simulation of [PERSON] oscillator. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multirow{2}{*}{**Initial**} & \multirow{2}{*}{**T**} & \multicolumn{4}{c}{**Step Size**} & \multirow{2}{*}{**Tolerance**} \\ \cline{3-3} \cline{5-6}  & & & **Min** & **Start** & **Max** & \\ \hline
4 & (1.6; 0; \(-\)0.1) & 15 & \(1\times 10^{-5}\) & \(5\times 10^{-3}\) & \(1\) & \(1\times 10^{-5}-1\times 10^{-9}\) \\
6 & (1.6; 0; \(-\)0.1) & 15 & \(1\times 10^{-5}\) & \(5\times 10^{-3}\) & \(1\) & \(1\times 10^{-7}\)–\(1\times 10^{-11}\) \\
8 & (1.6; 0; \(-\)0.1) & 15 & \(1\times 10^{-5}\) & \(5\times 10^{-3}\) & \(1\) & \(1\times 10^{-7}\)–\(-1\times 10^{-11}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter values used in simulation of the Rössler system. \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} & \(1\times 10^{-5}\) & 86 & 662 & 186 & 71 \\  & \(1\times 10^{-6}\) & 134 & 1423 & 328 & 100 \\  & \(1\times 10^{-7}\) & 210 & 3063 & 581 & 151 \\  & \(1\times 10^{-8}\) & 330 & 6598 & 1031 & 231 \\  & \(1\times 10^{-9}\) & 522 & 14,213 & 1831 & 359 \\ \hline \multirow{4}{*}{6} & \(1\times 10^{-7}\) & 146 & 191 & 307 & 138 \\  & \(1\times 10^{-8}\) & 199 & 301 & 484 & 177 \\  & \(1\times 10^{-9}\) & 273 & 475 & 766 & 219 \\  & \(1\times 10^{-10}\) & 377 & 751 & 1212 & 296 \\  & \(1\times 10^{-11}\) & 524 & 1188 & 1920 & 405 \\ \hline \multirow{4}{*}{8} & \(1\times 10^{-7}\) & 61 & 118 & 116 & 55 \\  & \(1\times 10^{-8}\) & 68 & 158 & 168 & 67 \\ \cline{1-1}  & \(1\times 10^{-9}\) & 84 & 217 & 244 & 83 \\ \cline{1-1}  & \(1\times 10^{-10}\) & 107 & 299 & 357 & 104 \\ \cline{1-1}  & \(1\times 10^{-11}\) & 137 & 414 & 522 & 132 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dependence between the total number of steps and tolerance while solving Rössler problem. A semi-explicit midpoint algorithm (8) for the Rossler system is as follows:  
\[ \begin{array}{l}z^{n+0.5}=z^{n}+\frac{h}{2}(b+z^{n}x^{n}-cz^{n}),\\ y^{n+0.5}=y^{n}+\frac{h}{2}(x^{n}+ay^{n}),\\ x^{n+\frac{1}{2}}=x^{n}+\frac{h}{2}(-y^{n+0.5}-z^{n+0.5}),\\ x^{n+1}=x^{n}+h(-y^{n+0.5}-z^{n+0.5}),\\ y^{n+1}=y^{n}+h\big{(}x^{n+0.5}+ay^{n+0.5}\big{)},\\ z^{n+1}=z^{n}+h\big{(}b+z^{n+0.5}x^{n+0.5}-cz^{n+0.5}\big{)}.\end{array} \]  
Table 2 presents parameter values used in our experiments.