# 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
the required norm is found, satisfying \(\|\bar{B}\|_{X_{1}}:=\|X_{1}^{-1}\bar{B}X_{1}\|_{\infty}=\max\{1,\gamma\}=1\) and \(\|\bar{B}-\mathds{1}\mathbb{1}^{T}A\|_{X_{1}}=\gamma<1\) in (45). Using this norm in (40) and (41) for \(e^{Y}:=M_{11}^{-1}\tau^{Y}\), it is seen with (38) that  
\[e_{n}^{Y}=\sum_{k=0}^{n}\mathds{1}(\mathbb{1}^{T}A\tau_{n-k}^{Y})+\underbrace {\sum_{k=0}^{n}\big{(}\bar{B}-\mathds{1}\mathbb{1}^{T}A\big{)}^{k}\tau_{n-k}^{ Y}}_{=O(h^{s})},\ 0\leq n<N. \tag{49}\]  
Only for \(\epsilon_{N}^{Y}\) a slight modification is required and the factors in the second sum have to be replaced by \((\bar{B}_{N}-\mathds{1}\mathbb{1}^{T}A)(\bar{B}-\mathds{1}\mathbb{1}^{T}A)^{k-1}\) for \(k>1\) with norms still of size \(O(\gamma^{k})\). Now, in all cases the loss of one order in the first sum in (49) may be avoided if the leading \(O(h^{s})\)-term of \(\tau_{n-k}^{Y}\) is canceled in the product with the left eigenvector, i.e., if \(\mathbb{1}^{T}A\tau_{n-k}^{Y}=O(h^{s+1})\). An analogous argument may be applied to the second term \(\|M_{22}^{-1}\tau^{P}\|\) in (40). The adjoint stability matrix \(\bar{B}^{\mathsf{T}}=(BA^{-1})^{\mathsf{T}}\) possesses the same eigenvalues as \(\bar{B}\) and its leading eigenvectors are also known: \(\bar{B}^{\mathsf{T}}\mathds{1}=\mathbb{1}\) and \((A\mathds{1})^{\mathsf{T}}\bar{B}^{\mathsf{T}}=\mathbb{1}^{T}B^{\mathsf{T}}=(A \mathds{1})^{\mathsf{T}}\) by preconsistency and an analogous construction applies. Under the conditions corresponding to local orders \((s,s-1)\) the leading error terms in \(\tau_{n}^{Y}=\frac{1}{s! }h^{s}\eta_{s}\otimes y^{(s)}(t_{n})+O(h^{s+1})\) and \(\tau_{n}^{p}=\frac{1}{(s-1)! }\eta_{s-1}^{*}\otimes p^{(s-1)}(t_{n})+O(h^{s})\) are given by  
\[\eta_{s} =\mathbf{c}^{s}-A^{-1}B(\mathbf{c}-\mathds{1})^{s}-sA^{-1}K \mathbf{c}^{s-1}, \tag{50}\] \[\eta_{s-1}^{*} =\mathbf{c}^{s-1}-A^{-T}B^{\mathsf{T}}(\mathbf{c}+\mathds{1})^{s- 1}+(s-1)A^{-T}K\mathbf{c}^{s-2}. \tag{51}\]  
Considering now \((\mathbb{1}^{T}A)\eta_{s}\) in (49) and similarly \((A\mathds{1})^{\mathsf{T}}\eta_{s-1}^{*}\) the following result is obtained. Theorem 3.1: _Let the Peer triplet with \(s>1\) stages satisfy the order conditions collected in Table 1 and let the solutions satisfy \(y\in\mathbb{C}^{s+1}[0,T]\), \(p\in\mathbb{C}^{s}[0,T]\). Let the coefficients of the standard Peer method satisfy the conditions_  
\[\mathds{1}^{\mathsf{T}}\big{(}A\mathbf{c}^{s}-B(\mathbf{c}- \mathds{1})^{s}-sK\mathbf{c}^{s-1}\big{)}= 0, \tag{52}\] \[\mathds{1}^{\mathsf{T}}\big{(}A^{\mathsf{T}}\mathbf{c}^{s-1}-B^{ \mathsf{T}}(\mathbf{c}+\mathds{1})^{s-1}+(s-1)K\mathbf{c}^{s-2}\big{)}= 0, \tag{53}\]  
_and let (44) be satisfied. Assume, that a Peer solution \((Y^{\mathsf{T}},P^{\mathsf{T}})^{\mathsf{T}}\) exists and that \(f\) and \(C\) have bounded second derivatives. Then, for stepsizes \(h\leq h_{0}\) the error of these solutions is bounded by_  
\[\|Y_{nj}-y(t_{nj})\|_{\infty}+h\|P_{nj}-p(t_{nj})\|_{\infty}=O(h^{s}), \tag{54}\]  
\(n=0,\ldots,N\)_, \(j=1,\ldots,s\)._  
Proof: Under condition (52) the representation (49) shows that \(\|M_{11}^{-1}\tau^{Y}\|=O(h^{s})\). In the same way follows \(\|M_{22}^{-1}\tau^{P}\|=O(h^{s-1})\) from condition (53). In (40) this leads to a common error \(\|\bar{Z}\|=O(h^{s-1})\) which may be refined for the state variable with (41) to \(\|\bar{Y}\|=O(h^{s})\). Remark 2: _The estimate (49) shows that superconvergence may be a fragile property and may be impaired if \(|\lambda_{2}|\) is too close to one, leading to very large values in the bound_  
\[\sum_{k=0}^{n}\|\big{(}\bar{B}-\mathds{1}\mathbb{1}^{\mathsf{T}}A\big{)}^{k} \|_{X_{1}}\leq\frac{1}{1-\gamma}\]  
_for the second term in (49). In fact, numerical tests showed that the value \(\gamma\cong|\lambda_{2}|\) plays a crucial role. While \(|\lambda_{2}|\doteq 0.9\) was appropriate for the three-stage method AP3o32f which shows order 3 in the tests in Section 3, for two four-stage methods with \(|\lambda_{2}|\doteq 0.9\) superconvergence was not seen in any of our three test problems below.