Piecewise-constant finite-volume state and face-flux reconstructions

Use the cells, time levels, states and interface fluxes from Conservative two-point finite-volume schemes for scalar laws. For $J_n^m=[t_n^m,t_{n+1}^m)$, the piecewise-constant state reconstruction and face-flux reconstruction are the measurable functions ${\mathcal{U}}_m(x,t)=U_j^{m,n},{\mathcal{Q}}_m(x,t)=Q_{j+1\mathrm{/}2}^{m,n}$ whenever $(x,t)\in I_j^m\times J_n^m$. Values on cell or time-grid boundaries may be assigned arbitrarily, since those boundaries have two-dimensional Lebesgue measure zero. In particular, for almost every $(x,t)$, ${\mathcal{Q}}_m(x,t)=F\text{\hspace{0.17em}⁣}({\mathcal{U}}_m(x,t),{\mathcal{U}}_m(x+h_m,t))\mathrm{.}$

Strong $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ convergence of reconstructions means convergence in $L^1$ on every compact subset of $\mathbb{R}\times (0,T)$. It is an explicit hypothesis in the weak-limit theorem below; it is not produced by the reconstruction definition.