# New Step Size Control Algorithm for Semi-Implicit Composition ODE Solvers
## 3 Results
### Problem 1: Rossler System
Table 3 provides data on total step evaluations for every considered algorithm. We tested three methods, of orders 4, 6, and 8. A double-precision floating-point data type was used. To compare the computational cost versus obtained error ratio for investigated step size control algorithms, including OCDM, DCOM, Blanes' embedded estimator (BEE), and the proposed technique, we present performance plots for them in Figure 4. Figure 5 illustrates the step size behavior for various local truncation error estimators. As one can see, while increasing the accuracy order of the desired solution, we simultaneously increase the number of composition stages, which leads to a significant decrease in calculation speed per step, but at the same time, it increases the overall performance due to larger step sizes. One can see from Figures 4 and 5 that the proposed step size control technique shows a noticeable increase in speed and accuracy for all investigated schemes. One can see that OCDM and the proposed scheme result in almost similar step size curve amplitude while DCOM and B  
A semi-implicit CD algorithm for the Van der Pol oscillator can be written as follows:  
\[ \begin{array}{c}y^{n+0.5}=y^{n}+\frac{h}{2}(m(1-x^{n}x^{n})y^{n}-x^{n}),\\ x^{n+1}=x^{n}+hy^{n+0.5},\\ y^{n+1}=\frac{y^{n+0.5}-\frac{h}{2}x^{n+1}}{1-\frac{h}{2}m\left(1-x^{n+1}x^{n+1 }\right)}.\end{array} \]  
A semi-explicit midpoint algorithm for the Van der Pol equation is as follows:  
\[ \begin{array}{c}y^{n+0.5}=y^{n}+\frac{h}{2}(m(1-x^{n}x^{n})y^{n}-x^{n}),\\ x^{n+0.5}=x^{n}+\frac{h}{2}y^{n+0.5},\\ x^{n+1}=x^{n}+hy^{n+0.5},\\ y^{n+1}=y^{n}+h\big{(}m\big{(}1-x^{n+0.5}x^{n+0.5}\big{)}y^{n+0.5}-x^{n+0.5} \big{)}.\end{array} \]  
Figure 6 shows the performance plots for solving the Van der Pol equation through different adaptive step size composition solvers. Figure 7 illustrates the step size behavior for various local truncation error estimators. Parameter values used in experiments can be found in Table 4. Gathered data on total number of steps for every considered algorithm is presented in Table 5.  
Figure 4: Performance plots for Rössler system simulation using composition solvers of orders 4 (**a**), 6 (**b**), and 8 (**c**). Figure 5: Step size behavior for different controllers under investigation while simulating Rössler system. Figure 6: Performance plots for Van der Pol oscillator simulation by solvers of orders 4 (**a**), 6 (**b**), and 8 (**c**). Figure 7: Graphs of step size behavior for different controllers under investigation for Van der Pol oscillator. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multirow{2}{*}{**Tolerance**} & \multicolumn{5}{c}{**Total Number of Steps**} \\ \cline{3-6}  & & **OCDM** & **DCOM** & **BEE** & **ECDM** \\ \hline \multirow{4}{*}{4} & \(1\times 10^{-5}\) & 115 & 823 & 232 & 133 \\  & \(1\times 10^{-6}\) & 162 & 1760 & 395 & 187 \\ \cline{1-1}  & \(1\times 10^{-7}\) & 258 & 3790 & 687 & 376 \\ \cline{1-1}  & \(1\times 10^{-8}\) & 442 & 8039 & 1206 & 445 \\ \cline{1-1}  & \(1\times 10^{-9}\) & 777 & 14,159 & 2133 & 698 \\ \cline{1-1}  & \(1\times 10^{-10}\) & 1748 & 22,080 & 3795 & 1009 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Dependence between the total number of steps and tolerance for Van der Pol system.