Sectional forward Euler separates temporal, quadrature, and cutoff errors

Assume all hypotheses of Sectional size discretization separates grid and cutoff errors, with continuous-model solution ${\mu }_t$ on $[0,T]$, initial mass $M_1$, grid spacing $\mathrm{\Delta }\le 1$, and pivot endpoint $R$. Let the finite grid have $m$ retained bins, take $T>0$, an integer $N\ge 1$, and $h=T\mathrm{/}N$. Initialize both the exact finite sectional system and forward Euler by that theorem's same retained projection and tail-mass reservoir. Let $y^k$ be the successful forward-Euler states under the uniform positivity CFL, and let ${\nu }_t^R$ be the exact finite-sectional retained measure. There are constants $C_{E,T,R,\mathrm{\Delta }}$ and $C_T$ independent of $N$ such that $d_{BL}({\sum }_i{n}_i^N{\delta }_{x_i},{\mu }_T)\le \sqrt{m}{C}_{E,T,R,\mathrm{\Delta }}h+e^{C_{T}T}[C_{Q}T\mathrm{\Delta }+\frac{\mathrm{\Delta }}{2}{M}_0({\mu }_T)+\frac{M_1}{R}+\frac{2K_{\ast }{M}_1^{2}T}{aR}]\mathrm{.}$ The first term is temporal error for the fixed finite ODE; the terms in $\mathrm{\Delta }$ and $R^{-1}$ are the separate grid and cutoff errors. The temporal constant may depend on the fixed grid and cutoff.