Conservative finite-volume strong-limit theorem

Fix $\phi \in C_c^{\mathrm{\infty }}(\mathbb{R}\times (-\mathrm{\infty },T))$. Its time support on $[0,T)$ is contained in $[0,T_{\phi }]$ for some $T_{\phi }<T$. For all sufficiently large $m$, choose $N_m\ge 3$ with $T_{\phi }<t_{N_m-2}^m<t_{N_m-1}^m<t_{N_m}^m<T$. By [F1] and [F2] these are precisely the scheme states and face fluxes of the reconstruction, so [F3] gives the exact summed update identity for this test.

The initial term in [F3] converges to the desired boundary term. Indeed, $h_m{U}_j^{m,0}={\int }_{I_j^m}{u}_0$, while ${\mathrm{\Phi }}_j^{m,0}$ is the average of $\phi$ on $I_j^m\times [0,{\tau }_m)$. Smoothness gives $\mathrm{\mid }{\mathrm{\Phi }}_j^{m,0}-\phi (x,0)\mathrm{\mid }\le C_{\phi }(h_m+{\tau }_m)$ for $x\in I_j^m$; summing the resulting errors over the finitely many cells meeting the support bounds the total by $C_{\phi }(h_m+{\tau }_m){\int }_K\mathrm{\mid }{u}_0\mathrm{\mid }\to 0$ on a fixed compact $K$.

For $n\ge 1$, the difference ${\mathrm{\Phi }}_j^{m,n}-{\mathrm{\Phi }}_j^{m,n-1}$ differs from $h_m^{-1}{\int }_{I_j^m}{\int }_{J_n^m}{\phi }_t$ by at most $C_{\phi }{\tau }_m^2$, by Taylor's formula in time for the smooth function $t\mapsto h_m^{-1}{\int }_{I_j^m}\phi (x,t)dx$. After multiplication by $h_m{U}_j^{m,n}$ and summation over $O(h_m^{-1})$ cells and $O({\tau }_m^{-1})$ time slabs in the fixed support, the total error is $O({\tau }_m)$. The missing first-slab integral is also $O({\tau }_m)$ because $\mathrm{\mid }{\mathcal{U}}_m\mathrm{\mid }$ is uniformly bounded on the compact support and ${\phi }_t$ is bounded. Hence the time sum in [F3] differs from ${\int }_0^T\int {\mathcal{U}}_m{\phi }_t$ by a quantity tending to zero.

For each $j,n$, translation of a cell interval and the fundamental theorem of calculus give ${\tau }_m({\mathrm{\Phi }}_{j+1}^{m,n}-{\mathrm{\Phi }}_j^{m,n})={\int }_{J_n^m}{\int }_{I_j^m}\frac{1}{h_m}{\int }_0^{h_m}{\phi }_x(x+s,t)dsdxdt$. Its difference from ${\int }_{J_n^m}{\int }_{I_j^m}{\phi }_x$ is at most $C_{\phi }{h}_m^2{\tau }_m$, by the bound on ${\phi }_{xx}$. The numerical face fluxes are uniformly bounded on cells meeting the test support because $F$ is continuous on the compact state square supplied by local boundedness. Summing over the finitely many cells and time slabs meeting the test support gives total error $O(h_m)$. Thus the flux sum in [F3] differs from ${\int }_0^T\int {\mathcal{Q}}_m{\phi }_x$ by a quantity tending to zero.

Strong local $L^1$ convergence gives $\int {\mathcal{U}}_m{\phi }_t\to \int u{\phi }_t$ on times bounded away from zero. On $I\times (0,ϵ)$, the local uniform bound for both ${\mathcal{U}}_m$ and $u$ makes the integral difference at most a constant times $\mathrm{\mid }I\mathrm{\mid }ϵ$; letting $ϵ\downarrow 0$ proves convergence on the full test support. By [F4], $\int {\mathcal{Q}}_m{\phi }_x\to \int f(u){\phi }_x$ on that support. Step 2.1 gives the initial term limit, and steps 2.2–2.3 identify the limits of the discrete sums. Taking the limit in [F3] proves the weak identity. By [F4], $u$ is locally essentially bounded, and [F5] identifies it as the stated local weak solution.