Differentiated explicit RK stages give local C1 order

Set $\xi =(x,\theta )$. The exact local solution has $V(t)=D_{\xi }u(t)$ because $V(t_n)=E$ and its displayed equation is the variational equation, with $D_{\xi }\theta =E_{\theta }$. Differentiating the cost integral gives $g(t)=D_{\xi }C(t{)}^T$. Thus at the local endpoint, $V^{+}=[D_x{\mathrm{\Phi }}_n{D}_{\theta }{\mathrm{\Phi }}_n],g^{+}=\left(\begin{array}{c}{D}_x{K}_n^T\\ D_{\theta }{K}_n^T\end{array}\right)\mathrm{.}$ The flow result and bounded continuous derivatives justify these differentiations uniformly.

Differentiate each actual stage, in its explicit order. If $V_i=D_{\xi }{Y}_i$, then $V_i=E+h{\sum }_{j<i}{a}_{ij}(f_{u,j}{V}_j+f_{\theta ,j}{E}_{\theta }),$ $D_{\xi }{\mathrm{\Psi }}_n=E+h{\sum }_i{b}_i(f_{u,i}{V}_i+f_{\theta ,i}{E}_{\theta }),$ $D_{\xi }{L}_n^T=h{\sum }_i{b}_i(V_i^T{\mathrm{\ell }}_{u,i}^T+E_{\theta }^T{\mathrm{\ell }}_{\theta ,i}^T)\mathrm{.}$ These are exactly the $V$ and $g$ outputs obtained by applying the same RK tableau to the augmented system. Its $C$ output is $L_n$ and its $u$ output is ${\mathrm{\Psi }}_n$. Consequently the assumed augmented local defect bounds ${\mathrm{\Psi }}_n-{\mathrm{\Phi }}_n$, $L_n-K_n$, and all four first-derivative differences by constants times $h^{p+1}$. Projection to a component does not increase the Euclidean norm; a matrix Frobenius bound also bounds its operator norm.

For a fixed finite tableau, the stage derivative recurrences in step 2.1 give uniform bounds on $D_x{Y}_i,D_{\theta }{Y}_i$ by induction, using $h\le h_0$ and the bounded first derivatives of $f$. Here $D_x{Y}_i=I+O(h)$ and $D_{\theta }{Y}_i=O(h)$. Substitution into the output derivatives gives $D_x{\mathrm{\Psi }}_n=I+O(h),D_{\theta }{\mathrm{\Psi }}_n=O(h),D_x{L}_n^T=O(h),D_{\theta }{L}_n^T=O(h),$ uniformly. In particular $\mathrm{\parallel }{D}_x{\mathrm{\Psi }}_n\mathrm{\parallel }\le 1+L_{h}h$ for a fixed constant $L_h$. The exact variational integral equation gives the same bounds for the exact maps, with $\mathrm{\parallel }{D}_x{\mathrm{\Phi }}_n\mathrm{\parallel }\le e^{Kh}\le 1+K{e}^{K{h}_0}h$. Enlarging the constants gives the common $L,M$ required by the preceding lemma.

For $p\le 4$, apply [F3] to the displayed autonomous augmented field itself. Its derivatives through order four exist and are uniformly bounded under the asserted augmented $C^4$ hypothesis. Joint base $C^5$ regularity is sufficient because its variational components use first derivatives of the base fields. All its exact paths and stages are admitted by [A1]. The source proposition therefore supplies the exact local augmented-defect premise used in step 2.1. This application concerns the differentiated augmented method, rather than just its state projection.

The exact maps have the required second-derivative bounds of size $O(h)$. To justify differentiating their variational equations, apply the common-tube $C^1$ flow result to the first variational system: its field is $C^1$ since $f$ is $C^2$, its initial derivative seed is constant, and its variational components remain bounded by [F2]. For coordinate directions $a,b$ of $\xi$, put $W_{ab}=D_{{\xi }_a}{D}_{{\xi }_b}u$. Then $W_{ab}^{\mathrm{\prime }}=f_u{W}_{ab}+D_{(u,\theta )}^{2}f[(V_a,E_{\theta ,a}),(V_b,E_{\theta ,b})],W_{ab}(t_n)=0.$ The forcing is uniformly bounded, so Grönwall on an interval of length $h$ gives $\mathrm{\parallel }{W}_{ab}\mathrm{\parallel }\le Ch$. Differentiating the cost integral twice produces the integrand $D_{(u,\theta )}^2\mathrm{\ell }[(V_a,E_{\theta ,a}),(V_b,E_{\theta ,b})]+{\mathrm{\ell }}_u{W}_{ab},$ which is uniformly bounded, and hence $D_{\xi }^2{K}_n=O(h)$. Thus $D_{\xi }^2{\mathrm{\Phi }}_n=O(h)$ and $D_{\xi }^2{K}_n=O(h)$. Applying the one-dimensional mean value inequality along each admitted state segment gives Lipschitz constants $Hh$ for $A_n^e,B_n^e,c_n^e,d_n^e$.

There is also a direct augmented global-error interpretation. Along a full trajectory append $C^{\mathrm{\prime }}=\mathrm{\ell }$ and the column $G^{\mathrm{\prime }}=S^T{\mathrm{\ell }}_u^T+{\mathrm{\ell }}_{\theta }^T$, with $C(0)=0,G(0)=0$ and $S^{\mathrm{\prime }}=f_{u}S+f_{\theta }$. Differentiating the actual stages shows that its numerical step is exactly $S^{+}=A_n^{h}S+B_n^h,C^{+}=C+L_n,G^{+}=G+S^T{c}_n^h+d_n^h\mathrm{.}$ The analogous exact step uses the exact maps. On bounded $S$ sets, step 2.1 therefore supplies a uniform $O(h^{p+1})$ local defect for $(u,S,C,G)$. Exact $S$ is bounded by Grönwall; numerical $S$ is bounded by the $1+Lh,Mh$ recurrence from step 3.1. The sums defining $C,G$ then bound them uniformly as well. Their stage values are bounded by the same finite-stage induction.

Steps 2.1–4.1 establish precisely the local hypotheses of the preceding lemma. That lemma now proves the global state, actual graph-tangent, backward nodal-adjoint and gradient bounds. In particular the adjoint conclusion is derived; it is not inserted as an additional hypothesis.

On a common compact convex augmented tube containing these values and joining stage segments, the $C^1$ augmented field has a derivative bound $K$. For two starting augmented states with difference $\delta$, explicit stage differences obey ${\delta }_i\le \delta +hK{\sum }_{j<i}\mathrm{\mid }{a}_{ij}\mathrm{\mid }{\delta }_j$. Induction gives ${\delta }_i\le D_i\delta$, where $D_1=1$ and $D_i=1+h_{0}K{\sum }_{j<i}\mathrm{\mid }{a}_{ij}\mathrm{\mid }{D}_j$. The output difference is at most $(1+hK{\sum }_i\mathrm{\mid }{b}_i\mathrm{\mid }{D}_i)\delta$. Thus the actual augmented method has one-step stability $1+\mathrm{\Lambda }h$. The global-error supplier applies, giving order-$p$ error in the actual tangent and accumulated cost derivative as well. This independently exhibits the augmented local-to-global route; step 5.1 additionally supplies the nodal adjoint.

For the displayed RK4 tableau the eight coefficient sums, in the order stated in the source proposition, are $1,1\mathrm{/}2,1\mathrm{/}3,1\mathrm{/}6,1\mathrm{/}4,1\mathrm{/}12,1\mathrm{/}8,1\mathrm{/}24$. For example $Ac=(0,0,1\mathrm{/}4,1\mathrm{/}2)$, $A{c}^{[2]}=(0,0,1\mathrm{/}8,1\mathrm{/}4)$, and $A^{2}c=(0,0,0,1\mathrm{/}4)$, which verify the mixed sums directly. Hence step 3.2 applies with $p=4$. No division by an individual $b_i$ or internal-stage adjoint order condition occurs in this argument.