Fourth-order global error for RK4 on a regularized vortex particle tube

Fix $\epsilon >0$, finitely many vortex weights, $T>0$, and the regularized finite-particle vector field of Equal-probability mean-field and weighted signed vortex particle systems, possibly with an added common smooth force correction $g_q(t,X,x_i)$ for spatial force quadrature. Suppose the resulting nonautonomous state field $f(t,X)$ is jointly $C^4$ on an open time-state domain, with derivatives through order four bounded on a common compact convex tube $\mathcal{T}$ compactly contained in that domain. Assume the exact path, RK4 grid states, and every RK4 stage lie in that tube for $h=T\mathrm{/}N\le h_0$, and the common force correction is included in these bounds. Let $C_4$ be a uniform exact-start defect constant, so the local defect is at most $C_4{h}^5$ as in One-step ODE methods, local defect, uniform order, and step stability. Let $L_{\ast }$ be the stability constant from One-step Lipschitz bound for classical RK4 on an admitted tube, so the one-step factor is $1+L_{\ast }h$. For exact initialization, the state errors $E_k$ satisfy ${\max }_{0\le k\le N}{E}_k\le \{\begin{array}{ll}{C}_4{\displaystyle \frac{e^{L_{\ast }T}-1}{L_{\ast }}}{h}^4,& L_{\ast }>0,\\ C_{4}T{h}^4,& L_{\ast }=0.\end{array}$ The tube and stage-admissibility assumptions are conditional; no invariance or global numerical-domain claim is made. The constant $C_4$ may depend on $\epsilon$, particle count, the quadrature field, and the tube.

For every step $k$, assume all RK4 stages generated from both the exact start $x(t_k)$ and the numerical start $X_k$ lie in $\mathcal{T}$, and both one-step maps are defined. Let $L$ be a uniform operator-norm bound for $D_{X}f$ on $\mathcal{T}$ and set $L_{\ast }=L{e}^{L{h}_0}$, as in the stability lemma. The only admitted start pair used in the stability estimate at step $k$ is $(x(t_k),X_k)$.