Conservative two-point finite-volume schemes for scalar laws

Let $T>0$, let $f\text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$ be continuous, and let $u_0\in L^{\mathrm{\infty }}(\mathbb{R})\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathbb{R})$ be initial data for $u_t+f(u{)}_x=0$ in the convention of Scalar conservation laws, fluxes and Cauchy data. For each $m\in \mathbb{N}$, choose $h_m>0$ and ${\tau }_m>0$ with $h_m\to 0$ and ${\tau }_m\to 0$. Define the uniform cells $I_j^m=[j{h}_m,(j+1)h_m)$ for $j\in \mathbb{Z}$ and the time levels $t_n^m=n{\tau }_m$ for $n\in {\mathbb{N}}_0$.

A continuous consistent numerical flux is a continuous map $F\text{\hspace{0.17em}⁣}:{\mathbb{R}}^2\to \mathbb{R}$ such that $F(a,a)=f(a)$ for every $a\in \mathbb{R}$. The associated conservative two-point finite-volume scheme starts from the cell averages $U_j^{m,0}=\frac{1}{h_m}{\int }_{I_j^m}{u}_0(x)dx$ and uses one shared interface flux $Q_{j+1\mathrm{/}2}^{m,n}=F(U_j^{m,n},U_{j+1}^{m,n})$ at every face. Its explicit update is $U_j^{m,n+1}=U_j^{m,n}-\frac{{\tau }_m}{h_m}(Q_{j+1\mathrm{/}2}^{m,n}-Q_{j-1\mathrm{/}2}^{m,n}),j\in \mathbb{Z},n\in {\mathbb{N}}_0\mathrm{.}$ Each internal face flux enters the two adjacent cell updates with opposite signs; this shared-flux difference is the meaning of conservative here. This definition imposes no CFL condition and no relation between ${\tau }_m$ and $h_m$.

The indexing convention is shown below. Each rectangle is one spatial cell; the arrows indicate the signed numerical flux, with positive sign oriented to the right. The same $Q_{j+1\mathrm{/}2}^{m,n}$ leaves cell $j$ and enters cell $j+1$.

The weak equation is one-dimensional on the full line. Compactly supported space-time tests therefore encounter only finitely many cells and no physical boundary flux. A boundedness or compactness estimate for the scheme is not part of this definition.