A fixed finite modified flow matches its method map to the stated Taylor order

If $K=\mathrm{\varnothing }$, its pointwise error family is empty, so every nonnegative real bounds it and $0$ is the least such bound. Thus $E_K(h)=0$ satisfies the estimate, for example with $h_0=C_N=1$. Assume henceforth $K\mathrm{\ne }\mathrm{\varnothing }$. For each admitted fixed $h$, [A1] and [F2] make the method and time-$h$ flow continuous in their initial state. By [F4] their error norm attains a finite nonnegative maximum on $K$; its maximum is its least nonnegative upper bound and its ordinary real supremum.

The method is a finite composition of smooth shears, or the smooth implicit branch in [A1]. For sufficiently small $h_0\le 1$, its derivatives with respect to the step parameter through order $N+2$ are bounded on the compact admitted comparison set. Applying [F3] to each component therefore gives ${\mathrm{\Phi }}_h(z)=z+{\sum }_{q=1}^{N+1}{h}^q{d}_q(z)+R_{\mathrm{\Phi }}(h,z)$ with $\mathrm{\parallel }{R}_{\mathrm{\Phi }}(h,z)\mathrm{\parallel }\le C_{\mathrm{\Phi }}\mathrm{\mid }h{\mathrm{\mid }}^{N+2}$ uniformly on $K$. The coefficients $d_q$ are those in [F1].

Hold $h$ fixed and put $X_h=X_H+{\sum }_{j=r}^N{h}^j{X}_j$. Along its trajectory $u_h(t,z)$, the integral equation in [F2] gives ${\mathrm{\partial }}_t{u}_h=X_h(u_h)$. Repeated differentiation in $t$ gives ${\mathrm{\partial }}_t^q{u}_h=B_q(h,u_h)$, where $B_1(h,w)=X_h(w)$ and $B_{q+1}(h,w)=D_w{B}_q(h,w)X_h(w)$. Each $B_q$ is a finite polynomial in $h$ with smooth coefficient functions of $w$, because $X_h$ is such a polynomial by [F1]. For $\mathrm{\mid }h\mathrm{\mid }\le h_0\le 1$ these coefficient functions and the required derivatives are uniformly bounded on the compact tube $K^{\mathrm{\prime }}$ by [F4]. Thus the time derivatives through $q=N+2$ are uniformly bounded on the stipulated trajectories. Apply [F3] in the time variable, with parameter $h$ fixed, between $t=0$ and $t=h$ (also for negative $h$). It gives ${\phi }_h^{[N]}(z)=z+{\sum }_{q=1}^{N+1}{h}^q{B}_q(h,z)\mathrm{/}q!+R_u(h,z)$, where $\mathrm{\parallel }{R}_u(h,z)\mathrm{\parallel }\le C_u\mathrm{\mid }h{\mathrm{\mid }}^{N+2}$ uniformly on $K$. No derivative of the flow with respect to the field parameter $h$ is used.

Expand the finitely many polynomials $B_q(h,z)$ in step 1.3 and collect equal powers of $h$. Every term of degree at least $N+2$ is bounded by a fixed constant times $\mathrm{\mid }h{\mathrm{\mid }}^{N+2}$ for $\mathrm{\mid }h\mathrm{\mid }\le 1$, because there are finitely many such terms and their coefficient functions are bounded on the compact tube. The recursion in [F1] (its coefficient construction) identifies all collected coefficients through degree $N+1$ with the method coefficients $d_q$ in step 1.2. Subtracting the two expansions leaves only their uniform remainders and the discarded higher-degree polynomial terms. Consequently $\mathrm{\parallel }{\mathrm{\Phi }}_h(z)-{\phi }_h^{[N]}(z)\mathrm{\parallel }\le C_N\mathrm{\mid }h{\mathrm{\mid }}^{N+2}$ for all $z\in K$, after increasing $C_N$ to a positive constant. Taking the attained maximum from step 1.1 proves the asserted estimate; step 1.1 already covers empty $K$.