Strong entropy convergence of monotone finite-volume schemes

Start with any subsequence of meshes. By [F4] it has a further subsequence along which $W_m$ converges in $C([0,T];L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1)$ to a map $u(t)$ with $u(0)=u_0$, values in $I$, and ${\mathcal{U}}_m\to u$ strongly in space-time $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$. The range condition gives the uniform local state bound required below.

Fix $k\in \mathbb{R}$ and a nonnegative $\phi \in C_c^{\mathrm{\infty }}(\mathbb{R}\times (0,T))$. Let ${\mathrm{\Phi }}_j^n$ be the average of $\phi$ on $I_j\times [t_n,t_{n+1}]$, and choose $N$ so that $\phi$ vanishes on and after the last time slab. Multiply the one-step inequality [F9] by $h{\mathrm{\Phi }}_j^n$, sum over the finitely many relevant $j,n$, telescope in time, and reindex the shared entropy fluxes in space. The resulting exact inequality is ${\sum }_{n=1}^{N-1}{\sum }_{j}h\mathrm{\mid }{U}_j^n-k\mathrm{\mid }({\mathrm{\Phi }}_j^n-{\mathrm{\Phi }}_j^{n-1})+{\sum }_{n=0}^{N-1}{\sum }_j\tau G_k(U_j^n,U_{j+1}^n)({\mathrm{\Phi }}_{j+1}^n-{\mathrm{\Phi }}_j^n)\ge 0.$ There is no initial term because $\phi$ is supported away from $t=0$.

The hypotheses of [F5] hold along this further subsequence: $f$ is continuous, the cell data and shared-flux recurrence are those of [F1], the mesh widths vanish, the states are bounded by $I$, and the strong local convergence is from step 1.1. Therefore $u$ satisfies the distributional weak equation with the prescribed initial boundary term.

On the compact state interval $J$ spanned by $I\cup \{k\}$, local Lipschitz continuity of $F$ gives a finite constant $C_k$ for which $\mathrm{\mid }{G}_k(a,b)-G_k(a,a)\mathrm{\mid }\le C_k\mathrm{\mid }b-a\mathrm{\mid }$. By [F3], the local space-time $L^1$ difference between the face entropy flux $G_k(U_j^n,U_{j+1}^n)$ and $q_k(U_j^n)=G_k(U_j^n,U_j^n)$ is at most $C_{k}T{h}_m\mathrm{TV}(u_0)$, which tends to zero. Also $q_k$ is Lipschitz on $I$ because $f\in C^1$, so strong convergence in step 1.1 gives $q_k({\mathcal{U}}_m)\to q_k(u)$ in local $L^1$.

The finite-sum expression in step 1.2 converges to ${\int }_0^T\text{\hspace{0.17em}⁣}{\int }_{\mathbb{R}}(\mathrm{\mid }u-k\mathrm{\mid }{\phi }_t+q_k(u){\phi }_x)dxdt\mathrm{.}$ Writing $\overline{{\phi }_t}{}_j^n$ for the average of ${\phi }_t$ on $I_j\times [t_n,t_{n+1}]$, Taylor's formula in time gives $\mathrm{\mid }{\mathrm{\Phi }}_j^n-{\mathrm{\Phi }}_j^{n-1}-{\tau }_m\overline{{\phi }_t}{}_j^n\mathrm{\mid }\le C_{\phi }{\tau }_m^2$. After multiplication by $h_m\mathrm{\mid }{U}_j^n-k\mathrm{\mid }$ and summing the $O(h_m^{-1})$ cells and $O({\tau }_m^{-1})$ slabs in the compact support, the error is $O({\tau }_m)$. For the flux term, translation of adjacent cell averages gives ${\tau }_m({\mathrm{\Phi }}_{j+1}^n-{\mathrm{\Phi }}_j^n)={\int }_{t_n}^{t_{n+1}}{\int }_{I_j}{h}_m^{-1}{\int }_0^{h_m}{\phi }_x(x+s,t)dsdxdt$; replacing the averaged derivative by ${\phi }_x$ has total error $O(h_m)$ because ${\phi }_{xx}$ and the numerical entropy fluxes are uniformly bounded on the compact support. The face-flux replacement in step 2.2 and strong state convergence supply the remaining limit. Passing to the limit in step 1.2 proves the Kruzhkov inequality for this $k,\phi$. Since both were arbitrary, all interior Kruzhkov inequalities hold.

The initial value and continuity from [F4] imply the strong local $L^1$ trace: for every compact interval $K$, $\mathrm{\parallel }u(t)-u_0{\mathrm{\parallel }}_{L^1(K)}\to 0$ as $t\downarrow 0$, hence the essential supremum over $0<t<\delta$ also tends to zero. Together with the entropy inequalities of step 3.1, [F6] makes $u$ a bounded Kruzhkov entropy solution.

Put $M_0:=\max \{\mathrm{\mid }{a}_{-}\mathrm{\mid },\mathrm{\mid }{a}_{+}\mathrm{\mid }\}$. By [F10], $f$ is Lipschitz on $[-M_0,M_0]$, which contains the range $I$ of every limit from step 4.1. Thus the range and flux hypotheses of [F7] hold for any pair of such limits. If $u_0\in L^1(\mathbb{R})$, [A2] and [F8] supply a bounded Kruzhkov solution, and [F7] makes it unique and identifies it with the limit of step 4.1. If $u_0\mathrm{\notin }{L}^1(\mathbb{R})$, step 4.1 itself supplies existence and [F7] supplies uniqueness. Thus in either branch there is a unique target entropy solution, without applying the global existence theorem outside its $L^1$ domain.

Every convergent subsequence from step 1.1 has an entropy-solution limit with the same initial data; by [F7] all such limits agree with the unique target from step 5.1 as space-time $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ classes. If the full sequence failed to converge strongly on some compact space-time set, a subsequence would stay at distance at least $\epsilon >0$ from that target; [F4] would give a further subsequence converging strongly there to an entropy solution, contradicting uniqueness. Therefore ${\mathcal{U}}_m\to u$ strongly in local space-time $L^1$. For the uniform-in-time conclusion, Fubini in [F6] turns this space-time equality into equality of the $L^1(K)$ sections for almost every $t$, for each compact interval $K$. The two local-$L^1$ limit maps are continuous, so they agree for every $t$: if their distance were positive at one time, continuity would make it positive on an interval of positive measure. Uniform-in-time local convergence of $W_m$ then follows by the same subsequence contradiction using [F4].