# New Step Size Control Algorithm for Semi-Implicit Composition ODE Solvers
## 2 Materials and Methods
### Embedded Error Estimators in Composition Schemes  
One way to evaluate a local truncation error is using pair of embedded methods, e.g., as in a well-known family of embedded Runge-Kutta schemes. Following the idea first described by [PERSON] et al. [9], one can use intermediate outputs obtained during the computation process of numerical approximation \(x_{n+1}=\Psi_{h}(x_{n})\approx x(t_{n+1})=x(t_{n}+h)\) with Equation (2). Using a composition scheme with \(s\) stages, one obtains \(s-1\) intermediate outputs:  
\[x_{n,0}=x_{n},\] \[x_{n,k}=\Psi_{h}(x_{n})=\Phi_{\gamma_{h}h}\circ\cdots\circ\Phi_{ \gamma_{1}h},\;k=1,\ldots,s-1.\]  
Approximation \(\widetilde{x}_{n+1}\) can be obtained as a linear combination of intermediate values \(x_{n,k}\) and further used for error estimation:  
\[\widetilde{x}_{n+1}=\sum\limits_{k=0}^{s-1}w_{k}x_{n,k}.\]  
Coefficients \(w_{k}\) were found by solving a system of linear equations, which represent order conditions [9]. The number of order conditions to be satisfied increases according to the desired order of \(\widetilde{x}_{n+1}\) solution:  
\[r_{0}\equiv w_{0}+\sum\limits_{k=1}^{s-1}w_{k}=1,\] \[r_{1}\equiv\sum\limits_{k=1}^{s-1}w_{k}\sum\limits_{j=1}^{k} \gamma_{j}=1,\] \[r_{2}\equiv\sum\limits_{k=1}^{s-1}w_{k}\left(\sum\limits_{j=1}^ {k}\gamma_{j}\right)^{2}=1,\] \[r_{3,1}\equiv\sum\limits_{k=1}^{s-1}w_{k}\left(\sum\limits_{j=1 }^{k}\gamma_{j}\right)^{3}=1,\] \[r_{3,2}\equiv\sum\limits_{k=1}^{s-1}w_{k}\sum\limits_{j=1}^{k} \gamma_{j}^{3}=1, \tag{9}\] \[r_{4,1}\equiv\sum\limits_{k=1}^{s-1}w_{k}\left(\sum\limits_{j=1 }^{k}\gamma_{j}\right)^{4}=1,\] \[r_{4,2}\equiv\sum\limits_{k=1}^{s-1}w_{k}\left(\sum\limits_{j=1 }^{k}\gamma_{j}^{4}+2\sum\limits_{j=1}^{k-1}\gamma_{j}^{3}\sum\limits_{l=j+1} ^{k}\gamma_{l}\right)=0,\] \[r_{4,3}\equiv\sum\limits_{k=1}^{s-1}w_{k}\left(\sum\limits_{j=1 }^{k}\gamma_{j}^{4}+2\sum\limits_{j=2}^{k}\gamma_{j}^{3}\sum\limits_{l=1}^{j-1 }\gamma_{l}\right)=0.\]  
To obtain a solution of accuracy order 2 from the fourth-order scheme, one must solve five equations from (9). The lowest number of stages required to satisfy order conditions in  
Figure 2: Scheme of DCOM algorithm. \(\gamma_{L},k=1,2,\ldots,s\) and \(\widetilde{\gamma}_{L},\;k=1,2,\ldots,u\) are coefficients for different composition schemes with \(s\) and \(u\) stages, respectively.  
this case equals 5, and a unique solution for the particular scheme of fourth order by [PERSON] et al. [7] from Table 1 (_s5 ord4_) will be as follows:  
\[\widetilde{x}_{n+1}=-x_{n}+w_{1}(x_{n,1}+x_{n,A})+w_{2}(x_{n,2}+x_{n,3}),\]  
with \(w_{1}=-1.40482876783862909\), \(w_{2}=2.40482876783863197\).  
To obtain a linear combination of order 4, one must solve the complete set of equations from (9). For the composition scheme with seven stages, a unique solution is as follows:  
\[\widetilde{x}_{n+1}=x_{n}+w_{1}(x_{n,1}-x_{n,6})+w_{2}(x_{n,2}-x_{n,5})+w_{3}( x_{n,3}-x_{n,4}).\]  
Following the same idea, one can find the required \(w_{i},\ i=1,2,3\) values for one of [PERSON]'s schemes [10] with composition coefficients from Table 1 (s7 ord6):  
\(w_{1}=-0.909832330075625028\),  
\(w_{2}=2.16331188722936796\),  
\(w_{3}=0.556955803872050015\).  
The proposed scheme can also be used with estimators of higher orders. In particular, one must solve a system of 13 linear equations to obtain an estimator of order 5 for a symmetric composition scheme with 17 stages of order 8 (s17 ord8). The estimator in this case reads  
\[\widetilde{x}_{n+1}=-x_{n}+\sum_{i=1}^{8}w_{i}(x_{n,i}-x_{n,17-i}),\]  
where  
\(w_{1}=-2.77811433347582461058\),  
\(w_{2}=1.43336350604816157334\),  
\(w_{3}=-2.35490307436226712937\),  
\(w_{4}=0.27249477875971647996\),  
\(w_{5}=3.09204406313073660493\),  
\(w_{6}=1.33511505989947708172\),  
\(w_{7}=0\),  
\(w_{8}=0\).  
Further, we will address this technique as Blanes' Error Estimation (BEE).