Discrete and continuous ODE gradients agree to order p under augmented error bounds

Let $T>0$, let $N\ge 1$ and $p\ge 1$ be integers, and set $h=T\mathrm{/}N$ and $t_n=nh$ for $0\le n\le N$. Let the finite graph defining $J_h$ have $N$ transitions with node states $x_0,\dots ,x_N$. Let $J$ and $J_h$ be the continuous and finite state-step objectives of The continuous ODE adjoint gives the parameter gradient with endpoint terms and The reverse recursion is the exact gradient of a finite state-step objective, respectively. Write $\lambda (t):=p(t)$ for the continuous adjoint in The continuous ODE adjoint gives the parameter gradient with endpoint terms, and write $x_n$ and ${\lambda }_n^h$ for the discrete state and adjoint. Set $u_n:=u(t_n)$ and ${\lambda }_n:=\lambda (t_n)$ for the exact state and adjoint values. Let $q(t,u,\lambda )=D_{\theta }\mathrm{\ell }(t,u,\theta {)}^T+D_{\theta }f(t,u,\theta {)}^T\lambda$ be the continuous running-gradient vector, evaluated exactly as $q(t):=q(t,u(t),\lambda (t))$. Suppose the exact and numerical state/adjoint pairs lie in a common tube and ${\max }_n(\mathrm{\parallel }{x}_n-u_n{\mathrm{\parallel }}_2+\mathrm{\parallel }{\lambda }_n^h-{\lambda }_n{\mathrm{\parallel }}_2)\le C_x{h}^p\mathrm{.}$ For the state part of this bound, it is sufficient that the hypotheses of Global error from uniform one-step defect and stability hold with a uniform $O(h^{p+1})$ exact-start defect; the coefficient conditions of Sufficient explicit Runge–Kutta order conditions through order four supply that defect for an explicit Runge--Kutta method of order $p\le 4$ under that item's $C^4$ and common-tube assumptions. No cited state-error result supplies the adjoint part, which is an independent hypothesis here. Write $b$ and $b_h$ for the column initial/terminal contribution vectors in the continuous and discrete gradient formulas. Assume the discrete running-gradient terms admit vectors ${\widehat{q}}_n$ and nonnegative weights $w_n$, $0\le n\le N$, with $\mathrm{\nabla }{J}_h=b_h+{\sum }_{n=0}^N{w}_n{\widehat{q}}_n,\mathrm{\nabla }J=b+{\int }_0^{T}q(t)dt,{\sum }_{n=0}^N{w}_n=T\mathrm{.}$ The weighted discrete representation is an explicit premise; it is not implied by the generic finite state-step adjoint recursion. Suppose $\mathrm{\parallel }{\widehat{q}}_n-q(t_n,x_n,{\lambda }_n^h){\mathrm{\parallel }}_2\le C_d{h}^p,\mathrm{\parallel }{b}_h-b{\mathrm{\parallel }}_2\le C_b{h}^p,$ and that the same weights give the quadrature $Q_h(q)={\sum }_{n=0}^N{w}_{n}q(t_n)$ satisfying ${\parallel Q_h(q)-{\int }_0^{T}q(t)dt\parallel }_2\le C_Q{h}^p\mathrm{.}$ If $q$ is Lipschitz in $(u,\lambda )$ on the common tube with constant $L_q$ in the sum norm, then $\mathrm{\parallel }\mathrm{\nabla }{J}_h-\mathrm{\nabla }J{\mathrm{\parallel }}_2\le (C_b+T(C_d+L_q{C}_x)+C_Q)h^p\mathrm{.}$ The adjoint, derivative-data and quadrature hypotheses are separate from state accuracy; without them the state estimate alone gives no gradient error conclusion.

There is also the following independent method-consistency conclusion. Under the local $C^1$ step-and-cost consistency, derivative stability, derivative Lipschitz, initial-data, terminal-derivative and tube hypotheses of Local C1 step consistency gives tangent, adjoint and gradient order, the actual numerical graph satisfies ${\max }_n\mathrm{\parallel }{D}_{\theta }{x}_n-D_{\theta }u(t_n){\mathrm{\parallel }}_2\le C_S{h}^p,{\max }_n\mathrm{\parallel }{\lambda }_n^h-\lambda (t_n){\mathrm{\parallel }}_2\le C_P{h}^p,$ $\mathrm{\parallel }\mathrm{\nabla }{J}_h-\mathrm{\nabla }J{\mathrm{\parallel }}_2\le C_G{h}^p,$ with that lemma's explicit constants, together with its state bound. These conclusions derive the global tangent and adjoint errors from local map errors. They do not assume the preceding global state/adjoint bound or weighted representation.

In particular, Differentiated explicit RK stages give local C1 order supplies these local hypotheses for the actual explicit RK state map and the cost using the same tableau, under its common-tube, base $C^2$, and local augmented-order hypotheses. For $p\le 4$, its augmented $C^4$ hypothesis and the existing coefficient conditions suffice; jointly $C^5$ base data are an explicit sufficient smoothness condition, and its displayed classical RK4 tableau gives a concrete order-four instance. Generic $p$ is retained through its genuine local augmented-order hypothesis. This branch permits negative or zero RK weights and makes no internal-stage adjoint accuracy claim.