A stage residual tolerance preserves collocation order

Fix $M$ nodes and an integer $1\le p\le M$. On a fixed interval use exact collocation stages $U_n^{\ast }$ and the collocation endpoint map ${\mathrm{\Psi }}_{h,n}$ of Collocation tableau from distinct quadrature nodes. Assume the hypotheses of Contraction and residual error bound for the SDC sweep hold uniformly, with contraction constants $q_n\le q_0<1$, the common field Lipschitz constant is $L$, and all exact and approximate stages and grid states stay in one compact tube. Assume also that the hypotheses of M-node collocation has local defect order M and Small-step collocation maps are Lipschitz stable hold uniformly at all mesh times for the exact and numerical starting states, including the bounded componentwise smoothness, exact-path derivative bound, and common branch and tube requirements. Let $U_n^{\ast }$ be the collocation stage vector for the numerical start $y_n$, and suppose the SDC stage vector $V_n$ belongs to its invariant product ball ${\mathcal{C}}_n$ and satisfies $\mathrm{\parallel }{S}_n(V_n)-V_n{\mathrm{\parallel }}_{\mathrm{\infty },2}\le (1-q_n)C_{\tau }{h}^{p+1}\mathrm{.}$ Starting from the exact initial value and setting $y_{n+1}=y_n+h{\sum }_j{b}_{j}f(t_n+c_{j}h,(V_n{)}_j)$, the inexact method has global error $O(h^p)$. Thus a fixed residual constant and a stage target $O(h^{p+1})$ preserve any collocation order $p\le M$; the bound is conditional on the stated local contraction and tube assumptions.