# Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
## 2 Materials and Methods
### Error Analysis
#### 2.2.3 Superconvergence of the Standard Method  
For \(s\)-stage Peer methods, global order \(s\) may be attained in many cases if other properties of the method have lower priority. For optimal stiff stability properties like A-stability, however, it may be necessary to sacrifice one order of consistency as in [20; 21]. Accordingly, in this paper the order conditions for the standard method are lowered by one compared to the requirements in the recent paper [13] to local orders \((s,s-1)\), see Table 1. Still, the higher global orders may be preserved to some extent by the concept of superconvergence which prevents the order reduction in the global error by cancellation of the leading error term. Superconvergence is essentially based on the observation that the powers of the forward stability matrix \(\bar{B}:=A^{-1}B\) may converge to a rank-one matrix which maps the leading error term of \(\tau^{Y}\) to zero. This is the case if the eigenvalue \(1\) of \(\bar{B}\) is isolated. Indeed, if the eigenvalues \(\lambda_{j}\), \(j=1,\ldots,s\), of the stability matrix \(\bar{B}\) satisfy  
\[1=\lambda_{1}>|\lambda_{2}|\geq\ldots|\lambda_{s}|, \tag{44}\]  
then its powers \(\bar{B}^{n}\) converge to the rank-one matrix \(\mathbf{1}\!\mathbf{1}^{\mathsf{T}}A\) since \(\mathbf{1}\!\mathbf{1}\) and \(\mathbf{1}\!\mathbf{1}^{\mathsf{T}}A\) are the right and left eigenvectors having unit inner product \(\mathbf{1}\!\mathbf{1}^{\mathsf{T}}A\mathbf{1}=1\) due to the preconsistency conditions \(A^{-1}B\mathbf{1}=\mathbf{1}\) and \(A^{-\mathsf{T}}B^{\mathsf{T}}\mathbf{1}=\mathbf{1}\) of the forward and backward standard Peer method, see (30) and (33). The convergence is geometric, i.e.,  
\[\|\bar{B}^{n}-\mathbf{1}\!\mathbf{1}^{\mathsf{T}}A\|=\|\left(\bar{B}-\mathbf{1 }\!\mathbf{1}^{\mathsf{T}}A\right)^{n}\|=O(\gamma^{n})\to 0,\,n\to\infty, \tag{45}\]  
for any \(\gamma\in(|\lambda_{2}|,1)\). Some care has to be taken here since the error estimate (40) depends on the existence of special norms satisfying \(\|\bar{B}\|_{\chi_{1}}:=\|X_{1}^{-1}\bar{B}X_{1}\|_{\infty}=1\), resp. \(\|\bar{B}^{\mathsf{T}}\|_{\chi_{2}}=1\). Concentrating on the forward error \(\|M_{11}^{-1}\tau^{Y}\|\), a first transformation of \(\bar{B}\) is considered with the matrix  
\[X= \begin{pmatrix}1&-\beta^{\mathsf{T}}\\ \mathbf{1}_{s-1}&I_{s-1}-\mathbf{1}\!\mathbf{1}_{s-1}\beta^{\mathsf{T}}\end{pmatrix} ,\,X^{-1}= \begin{pmatrix}\beta_{1}&\beta^{\mathsf{T}}\\ -\mathbf{1}_{s-1}&I_{s-1}\end{pmatrix} , \tag{46}\]  
where \((\beta_{1},\beta^{\mathsf{T}})=\mathbf{1}\!\mathbf{1}^{\mathsf{T}}A\). Since \(Xe_{1}=\mathbf{1}\!\mathbf{1}\) and \(e_{1}^{\mathsf{T}}X^{-1}=\mathbf{1}\!\mathbf{1}^{\mathsf{T}}A\), the matrix \(X^{-1}\bar{B}X\) is block-diagonal with the dominating eigenvalue \(1\) in the first diagonal entry. Due to (44) there exists an additional nonsingular matrix \(\Xi\in\mathbb{R}^{(s-1)\times(s-1)}\) such that the lower diagonal block of \(X^{-1}BX\) has norm smaller one, i.e. \[\|\Xi^{-1}\big{(}-\mathbf{1}\!\mathbf{1}_{s-1},I_{s-1}\big{)}\bar{B}\! \begin{pmatrix}-\beta^{\mathsf{T}}\\ I_{s-1}-\mathbf{1}\!\mathbf{1}_{s-1}\beta^{\mathsf{T}}\end{pmatrix} \Xi\|_{ \infty}=\gamma<1. \tag{47}\]  
Hence, with the matrix  
\[X_{1}:=X\! \begin{pmatrix}1&\\ &\Xi\end{pmatrix}  \tag{48}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Steps** & **Forward: \(\boldsymbol{q_{1}=s}\)** & **Adjoint: \(\boldsymbol{q_{2}=s-1}\)** \\ \hline Start, \(n=0\) & (29) & (33),(61),\(\beta=0\) \\ \hline Standard, \(1\leq n<N\) & (30) & (33) \\ Superconvergence & (52) & (53) \\ Compatibility & (67) & (67) \\ \hline Last step & (30), \(n=N\) & (33), \(n=N-1\) \\ End point & (31) & (34),(61),\(\beta=N\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Combined order conditions for the Peer triplet, including the compatibility condition (67) and the condition (61) for full matrices \(K_{N}\).