Finite-order modified energy control on a compact tube

Let $Q\subseteq {\mathbb{R}}^d$ be open and star-shaped about a point $q_{\ast }\in Q$, meaning $q_{\ast }+t(q-q_{\ast })\in Q$ for every $q\in Q$ and $t\in [0,1]$. Let $M$ be real symmetric positive-definite, $U=Q\times {\mathbb{R}}^d$, and $H(q,p)=\frac{1}{2}{p}^T{M}^{-1}p+V(q)$ with $V\in C^{\mathrm{\infty }}(Q)$. Choose one of this page's fixed-step methods: kick-then-drift symplectic Euler with $r=1$, or local leapfrog or an admitted implicit midpoint branch with $r=2$. Fix a finite truncation order $N\ge r$ and choose $h_0\in (0,1]$ small enough for the selected map and comparison flow branches. For one step size $0<h\le h_0$, let $z_{k+1}={\mathrm{\Phi }}_h(z_k)$ for $k=0,\dots ,n-1$, with $n\in {\mathbb{N}}_0$. Assume every numerical endpoint, every method stage, every truncated modified-flow comparison segment, and every segment used to compare $H$ with $H_h^{[N]}$ lies in one compact tube $K⋐U$. Then there are constants $C_0,C_1$, independent of $h,n$, such that $\mathrm{\mid }H(z_n)-H(z_0)\mathrm{\mid }\le 2C_0{h}^r+C_{1}n{h}^{N+1}\mathrm{.}$ The bound holds only while the stated compact-tube and domain hypotheses hold; if $nh\le T$, its accumulated defect term is at most $C_{1}T{h}^N$. It asserts neither exact conservation of $H$ by the numerical method nor convergence of an infinite modified series.

Symplecticity, reversibility, exact invariants, and modified-energy control are separate properties. This result concerns only the unconstrained maps specified above. Lie and Strang splitting are treated at the exact split-flow compositions ↗, while constrained integration is treated at the classical RATTLE step ↗; no claim here transfers to those methods.