Staged fixed-regularization PIC error budget

Apply [F1] to each unit source $\mu ={\delta }_y$. The explicit constants in [F1] depend on the source through $\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}$ and suprema of derivatives of its continuum potential and field. Here $\mathrm{\parallel }{\delta }_y{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}=1$, and those continuum functions for ${\delta }_y$ are translates of the functions for ${\delta }_0$; their derivative suprema are therefore independent of $y$. For the corrected shifted-product quadrature constant, translating the unit-source field by $y$ is absorbed by translating both its spatial variable and the free gather location in the joint supremum. That constant and the other shape quadrature constants therefore do not depend on the source translate. One set of constants $C_{\mathrm{s}\mathrm{r}\mathrm{c}}^{(1)},C_3^{(1)},C_{\mathrm{i}\mathrm{n}\mathrm{t}}^{(1)}$ gives the stated uniform phase-kernel defect bound, without asserting translation invariance of the finite grid kernel.

Work at a retained time $t=t_n=n\mathrm{\Delta }t\le T$ on the stated grid, whose step obeys $0<\mathrm{\Delta }t\le h_0$. Let $A$ be the finite support of ${\nu }_0^h$ and $I_a=\{p:z_p^h(0)=a\}$, $m_a=\mathrm{\mid }{I}_a\mathrm{\mid }$ for $a\in A$. For any prescribed coupling ${\pi }_0$ of ${\nu }_0^h$ and ${\mu }_0$, define a measure on the second coordinate for each label $p\in I_a$ by ${\lambda }_p(B)={\pi }_0(\{a\}\times B)\mathrm{/}{m}_a$. Since ${\nu }_0^h(\{a\})=m_a\mathrm{/}P$, each ${\lambda }_p$ has mass $1\mathrm{/}P$, and for each $a$ the sum over $p\in I_a$ is ${\pi }_0(\{a\}\times \cdot )$. By the duplicate-path argument in Periodic PIC particle and force errors under a prescribed initial coupling, all labels in $I_a$ share the same exact path $X_t(a)$. Thus summing these label measures gives exactly the pushforward ${\mathrm{\Pi }}_t=(X_t,Y_t{)}_{\mathrm{\#}}{\pi }_0$, the Dobrushin-transported coupling, where $Y_t$ is the continuum characteristic flow. In particular, ${\mathrm{\Pi }}_t$ has the exact grid and continuum marginals. Its cost is at most $e^{2L_{\epsilon }T}(D_0+T{\delta }_{\epsilon ,h})$ by [F2]. Replace $z_p^h(t)$ in each summand by the RK4 state $z_p^{\mathrm{\Delta }t}(t)$. The resulting measure is a coupling of ${\nu }_{P,h}^{\mathrm{\Delta }t}(t)$ and ${\mu }_{\epsilon }(t)$, and pointwise triangle inequality increases its cost by at most $P^{-1}{\sum }_p\mathrm{\mid }{z}_p^{\mathrm{\Delta }t}-z_p^h\mathrm{\mid }$. By Cauchy–Schwarz this is at most $P^{-1\mathrm{/}2}\mathrm{\parallel }{Z}^{\mathrm{\Delta }t}-Z^h{\mathrm{\parallel }}_2\le P^{-1\mathrm{/}2}{C}_{\mathrm{R}\mathrm{K}}\mathrm{\Delta }{t}^4$ using [F3]. This gives the claimed bound with $D_0$ in place of ${\delta }_0$.

The bound of step 1.2 holds for every prescribed initial coupling. Taking the infimum of $D_0$ gives ${\delta }_0$ by [F4], without selecting a minimizing plan. This proves the stated staged error budget.