Fourth-order RK4 trajectory error for the fixed-grid periodic PIC ODE

Fix $\epsilon >0$, grid width $h$, particle count $P\ge 1$, and final time $T>0$. Apply classical RK4 from Classical RK4 step for the fixed-grid regularized PIC particle ODE to the autonomous finite-dimensional PIC system $\dot{Z}=F_{\epsilon ,h}(Z),(F_{\epsilon ,h}(Z){)}_p=\frac{1}{P}{\sum }_{r=1}^P{\mathbb{K}}_{\epsilon ,h}(z_p,z_r),z_p=(X_p,V_p)\mathrm{.}$ Let $Z(t)$ be an exact solution on $[0,T]$. Assume $F_{\epsilon ,h}$ is $C^4$ on an open set containing a common compact convex tube $\mathcal{T}$; the exact path, all numerical states, and every RK4 stage generated from both the exact and numerical starting states lie in $\mathcal{T}$; and all derivatives of $F_{\epsilon ,h}$ through order four are bounded there. Choose a uniform step $0<h_t\le h_0$ with $h_0>0$, initialize exactly, and retain every time $t_k=k{h}_t\le T$ for integers $k\ge 0$; $T\mathrm{/}{h}_t$ need not be an integer. The stage hypotheses concern each step ending by $T$. Put $L_F=2L_{\epsilon ,h},L_{\ast }=L_F{e}^{L_F{h}_0},$ and choose $C_4$ so that the exact-start defect on this tube is at most $C_4{s}^5$ for admitted steps $0<s\le h_0$ ending by $T$. The local-order supplier gives such constants for a sufficiently small positive $h_0$ under the stated smoothness and tube assumptions. Then, for every grid time $t_k=k{h}_t\le T$, $\mathrm{\parallel }{Z}_k-Z(t_k){\mathrm{\parallel }}_2\le \{\begin{array}{ll}{C}_4{\displaystyle \frac{e^{L_{\ast }T}-1}{L_{\ast }}}{h}_t^4,& L_{\ast }>0,\\ C_{4}T{h}_t^4,& L_{\ast }=0.\end{array}$ The tube, exact-solution, and stage-admissibility hypotheses are conditional; this is a fixed-$\epsilon$, fixed-grid finite-particle time error, not a continuum or singular-force estimate.