A strong limit of hydrostatic HLL updates solves the smooth-bed balance law weakly

By [F5], applied to the nonnegative bulk and initial depths, the strong local convergences, $m=hu$, the dry-velocity convention and the local initial velocity bound, both $U$ and $U_0$ belong to $K$ almost everywhere. Thus $F(U)$ in the weak formulation is defined.

Fix $\phi$ and choose $N_k$ so that $\phi$ vanishes for $t\ge t_{N_k-1}^k$ for all sufficiently large $k$. Apply [F2] to the actual states, shared face fluxes and source terms of the scheme; its exact finite sum is zero for each such $k$.

The initial term in [F2] tends to ${\int }_{\mathbb{R}}{U}_0(x)\cdot \phi (x,0)dx$. Indeed, on each cell-time slab touching $t=0$, the averaged test differs from $\phi (x,0)$ by at most $C_{\phi }(d_k+{\tau }_k)$, and the uniform state bound including the initial layer makes the sum of these quadrature errors tend to zero on the fixed compact support. The initial reconstruction convergence then gives the claimed limit.

The time-difference sum in [F2] differs from ${\int }_0^T\int {\mathcal{U}}_k\cdot {\phi }_t$ by $O({\tau }_k)$. Taylor's formula in time makes each difference of adjacent cell-time test averages equal to the integral of ${\phi }_t$ over the current slab, up to $C_{\phi }{\tau }_k^2$ after spatial averaging; the uniform state bound and the $O(d_k^{-1}{\tau }_k^{-1})$ cells/slabs meeting the support give total error $O({\tau }_k)$. The omitted initial-slab integral is also $O({\tau }_k)$.

The face-flux sum in [F2] differs from ${\int }_0^T\int {\mathcal{F}}_k\cdot {\phi }_x$ by $O(d_k)$. For each cell, the difference of adjacent test averages is the spatial average of ${\int }_0^{d_k}{\phi }_x(x+s,t)ds$; replacing this average by ${\phi }_x(x,t)$ has error at most $C_{\phi }{d}_k^2{\tau }_k$ per cell-time slab. The local depth and velocity bounds give $h^{\ast }\le H$, $\mathrm{\mid }{u}^{\ast }\mathrm{\mid }\le M$, and wave speeds at most $M+\sqrt{gH}$. The physical fluxes and reconstructed states are bounded; in the middle HLL branch its equivalent convex-combination formula has coefficient $\alpha \beta \mathrm{/}(\alpha +\beta )\le \max (\alpha ,\beta )$, so the face fluxes are uniformly bounded as well. The number of cells/slabs meeting the support then gives the stated total error. The source sum in [F2] is exactly ${\int }_0^T\int {\mathcal{S}}_k\cdot \phi$ because ${\mathcal{S}}_k$ is constant on each slab.

Choose a compact spatial interval containing the test support and all cells meeting it for sufficiently large $k$, and choose $T_0<T$ beyond the temporal support. On this compact rectangle the hypotheses give uniform bounds for $h_k,u_k$ through $t=0$; the interior strong limits inherit the same essential bounds. Hence ${\mathcal{U}}_k$, $U$, $F(U)$ and the products with the fixed test derivatives are uniformly bounded. Step 1.5 supplies a uniform bound for ${\mathcal{F}}_k$. For sufficiently large $k$, $d_k\le 1$, and [F6] together with the local bounds on $b^{\mathrm{\prime }}$ and the depths gives a uniform bound for ${\mathcal{S}}_k$; the limit source $(0,-gh{b}_x)$ is bounded too. Thus the contributions of all state, flux and source integrals from the initial layer $[0,\epsilon ]$ are bounded by $C\epsilon$, uniformly in sufficiently large $k$ and for the limit integrands. On the remaining compact rectangle with $t\ge \epsilon$, the assumed strong convergence gives convergence of the state integral, [F4] gives convergence of the face-flux integral, and [F3] gives convergence of the source integral. Taking $k\to \mathrm{\infty }$ for fixed $\epsilon >0$ and then $\epsilon \downarrow 0$ proves convergence of these full integrals. Step 1.3 supplies the initial term, steps 1.4–1.5 identify the limits of the discrete sums, step 1.2 supplies the exact discrete identity, and step 1.1 its state admissibility. Passing to the limit proves the displayed weak formulation.