Piecewise-linear time interpolant of a finite-volume scheme

Let $U_j^n$ be the states on a mesh from Monotone finite-volume scheme and invariant-interval CFL, with cell-time slabs $I_j\times [t_n,t_{n+1})$ and $t_n=n\tau$. The piecewise-linear time interpolant $W_{h,\tau }$ is constant in $x$ on each cell and, for $t\in [t_n,t_{n+1})$, is $W_{h,\tau }(x,t):=(1-\theta )U_j^n+\theta U_j^{n+1},\theta :=\frac{t-t_n}{\tau },x\in I_j\mathrm{.}$ The piecewise-constant state ${\mathcal{U}}_{h,\tau }$ and face flux ${\mathcal{Q}}_{h,\tau }$ use the convention of Piecewise-constant finite-volume state and face-flux reconstructions. Values on mesh boundaries are irrelevant to the corresponding Lebesgue classes.

The interpolant is affine in time on each slab and has the scheme values at both time endpoints. Its purpose here is to obtain time-uniform local compactness and the strong initial trace; the weak-limit supplier is applied to ${\mathcal{U}}_{h,\tau }$ after proving that the two reconstructions have the same strong local limit.