# New Step Size Control Algorithm for Semi-Implicit Composition ODE Solvers
## 2 Materials and Methods  
### Using Variable-Step Integration with Composition Schemes  
The main idea of the composition scheme [5; 6] is to achieve an approximation of higher order by combining solutions obtained by a basic method \(\Phi_{h}\) as follows.  
Let us choose numbers \(\gamma_{1},\ldots,\gamma_{s}\) satisfying the consistency condition  
\[\gamma_{1}+\gamma_{2}+\cdots+\gamma_{s}=1. \tag{1}\]  
Using Equation (1), we can obtain a composition method with s stages:  
\[\Psi_{h}=\Phi_{\gamma_{s}h}\circ\cdots\circ\Phi_{\gamma_{2}h}\circ\Phi_{\gamma _{1}h}. \tag{2}\]  
Coefficients \(\gamma_{1},\ldots,\gamma_{s}\) are chosen to increase the order of the overall method \(\Psi_{h}\) over the order of the basic method. They can be found using various techniques. First, consider a [PERSON] sequence proposed by [PERSON] in 1990 [5] which has proven to be simple and efficient, and is widely known among scholars in the field. For a three-stage composition method, the coefficients are  
\[\gamma_{1}=\gamma_{3}=\left(2-2^{\frac{1}{p+1}}\right)^{-1},\gamma_{2}=-2^{ \frac{1}{p+1}}\left(2-2^{\frac{1}{p+1}}\right)^{-1}, \tag{3}\]  
where \(p\) is the accuracy order of the basic method \(\Phi_{h}\).  
Following the idea of [8], one can conclude that choosing the [PERSON] basic method of order 2 for the composition method with coefficients Equation (3) results in a three-stage method of order 4.  
The other several considered sequences of coefficients are based on [PERSON]'s [10] interpretation of [PERSON]'s scheme and are presented in Table 1. The notation _sNordM_ is used following the original work and corresponds to a scheme with N stages which gives an approximation of accuracy order M.  
\begin{table}
\begin{tabular}{c c} \hline \hline  & s\(\mathrm{lord2}\) \\ \hline \(\gamma_{1}\) & \(1\) \\ \hline  & s\(\mathrm{5 ord4}\) \\ \hline \(\gamma_{1}=\gamma_{2}=\gamma_{4}=\gamma_{5}\) & \(0.414490771794375737142354063\) \\ \(\gamma_{3}\) & \(-0.65796308717750294856941625\) \\ \hline  & s\(\mathrm{7 ord6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Coefficients for composition schemes of different orders of accuracy [10].  
Implementing the ODE solver with adaptive step size requires a formula for calculating a new step size value based on the previous one. The general formula for a new step size in a step control algorithm is as follows:  
\[h_{n+1}=h_{n}R,\]  
where \(R\) is a coefficient which contains information about computation error and can be obtained using different algorithms. In our study, we used the conventional step size control algorithm presented by [PERSON] et al. in [11], which states that after choosing an initial step size \(h_{n}\), two approximations of the solution are to be found using two integration methods with different accuracy orders to evaluate the local truncation error. Let us refer to these approximations as \(\mathbf{x}_{n}\) and \(\widetilde{\mathbf{x}}_{n}\). Using those approximations, one can find a new step size value \(h_{n+1}\) as follows:  
\[h_{n+1}=h_{n}\bigg{(}\frac{Tol}{\mathbf{x}_{n}-\widetilde{\mathbf{x}}_{n}} \bigg{)}^{\frac{1}{n+1}}, \tag{4}\]  
where \(m\) corresponds to the lowest accuracy order of the schemes, and \(Tol\) is the desired error tolerance.