Monotone finite-volume scheme and invariant-interval CFL

Let $f\in C^1(\mathbb{R})$ and let $u_0\in {\mathrm{B}\mathrm{V}}_b(\mathbb{R})$ as in Bounded BV data on the real line. Put $a_{-}:={\mathrm{ess\hspace{0.17em}inf}}_{\mathbb{R}}{u}_0$, $a_{+}:={\mathrm{ess\hspace{0.17em}sup}}_{\mathbb{R}}{u}_0$ and $I:=[a_{-},a_{+}]$. A monotone consistent two-point flux on $I$ is a locally Lipschitz map $F\text{\hspace{0.17em}⁣}:{\mathbb{R}}^2\to \mathbb{R}$ such that $F(s,s)=f(s)$ for every $s$, $F(\cdot ,b)$ is nondecreasing on $I$, and $F(a,\cdot )$ is nonincreasing on $I$. Choose finite coordinate Lipschitz bounds $L_1,L_2\ge 0$ on $I^2$: $\mathrm{\mid }F(a,b)-F(a^{\mathrm{\prime }},b)\mathrm{\mid }\le L_1\mathrm{\mid }a-a^{\mathrm{\prime }}\mathrm{\mid },\mathrm{\mid }F(a,b)-F(a,b^{\mathrm{\prime }})\mathrm{\mid }\le L_2\mathrm{\mid }b-b^{\mathrm{\prime }}\mathrm{\mid }\mathrm{.}$ Local Lipschitz continuity on the compact square $I^2$ supplies such bounds.

For a mesh width $h>0$ and time step $\tau >0$, use the half-open cells $I_j=[jh,(j+1)h)$, initial cell averages $U_j^0:=\frac{1}{h}{\int }_{I_j}{u}_0(x)dx,$ and the shared-flux update $U_j^{n+1}=U_j^n-\lambda (F(U_j^n,U_{j+1}^n)-F(U_{j-1}^n,U_j^n)),\lambda :=\frac{\tau }{h},j\in \mathbb{Z},n\in {\mathbb{N}}_0\mathrm{.}$ The full invariant-interval CFL bound is $\lambda (L_1+L_2)\le 1.$ The flux-difference scheme is conservative in the shared-face sense of Conservative two-point finite-volume schemes for scalar laws. The update stencil and positive rightward flux orientation are shown here; the proofs use the displayed recurrence, not the drawing.

diagram failed to render:
\begin{tikzpicture}[x=1.05cm,y=1cm]
  \draw (0,0) rectangle (2,0.7);
  \draw (2,0) rectangle (4,0.7);
  \draw (4,0) rectangle (6,0.7);
  \node at (1,0.35) {$U_{j-1}^n$};
  \node at (3,0.35) {$U_j^n$};
  \node at (5,0.35) {$U_{j+1}^n$};
  \draw[->] (1.55,1.05) -- (2.45,1.05);
  \draw[->] (3.55,1.05) -- (4.45,1.05);
  \node at (2,1.32) {$F(U_{j-1}^n,U_j^n)$};
  \node at (4,1.32) {$F(U_j^n,U_{j+1}^n)$};
  \node[align=center] at (3,-0.7) {$U_j^{n+1}=U_j^n-\lambda(\text{right flux}-\text{left flux})$};
\end{tikzpicture}

The same face value leaves one cell and enters the next with opposite signs. Monotonicity and the stated CFL bound make the three-argument update map order-preserving on $I^3$ and preserve constant states, as proved in Invariant range, TVD and time increments for a monotone finite-volume update.