= Solution
The <Engquist-Osher method> is an upwind <finite volume method>. For cell values $U_j^n$, mesh width $h$ and time step $k$, put $r=k/h$ and use the conservative update
$$
U_j^{n+1}=U_j^n-r\bigl[F(U_j^n,U_{j+1}^n)-F(U_{j-1}^n,U_j^n)\bigr],
$$
where the <numerical flux> is
$$
\boxed{F(a,b)=f(0)+\int_0^a\max\{f'(s),0\}ds+\int_0^b\min\{f'(s),0\}ds.}
$$
<Consistency of a numerical method> follows from $F(u,u)=f(u)$. Positive <characteristic speeds> use the left state, and negative speeds use the right state. The flux difference telescopes when summed over cells, giving a discrete <conservation law>. The method is first order in space and time for smooth solutions; its <conservation law> form is also appropriate when <shock waves> develop.
Let $s_*$ be the unique <sonic point of a scalar flux>, $f'(s_*)=0$. <Convexity> implies $f'<0$ to its left and $f'>0$ to its right. Put $f_*=f(s_*)$ and split
$$
f^+(u)=\begin{cases}0&u\le s_*,\\f(u)-f_*&u>s_*,\end{cases}
\qquad f^-(u)=\begin{cases}f(u)-f_*&u<s_*,\\0&u\ge s_* .\end{cases}
$$
Then $f=f_*+f^++f^-$, $(f^+)'\ge0$, $(f^-)'\le0$, and
$$
F(a,b)=f_*+f^+(a)+f^-(b).
$$
This is the <sonic-point splitting of a convex Engquist-Osher flux>. If both states are above $s_*$, $F=f(a)$; if both are below it, $F=f(b)$. Across a transsonic <rarefaction wave>, $a\le s_*\le b$, the flux is the minimum $f_*$. In the opposite crossing it is $f(a)+f(b)-f_*$, which in general differs from the exact <Godunov numerical flux>. This distinction does not affect the <monotonicity> proof.
Here <stability> requires the usual <Courant–Friedrichs–Lewy condition>. Let the initial values lie in $[m,M]$, and choose
$$
\boxed{r\max_{u\in[m,M]}|f'(u)|\le1.}
$$
A differentiable <convex function> has a continuous <derivative> on this interval, so the maximum is finite. Write the three-input update as $H(a,b,c)$. Differentiating gives
$$
H_a=r(f^+)'(a)\ge0,\qquad H_c=-r(f^-)'(c)\ge0,\qquad
H_b=1-r\bigl[(f^+)'(b)-(f^-)'(b)\bigr]=1-r|f'(b)|\ge0.
$$
Thus the update $T$ preserves componentwise order. Constants are fixed, since $H(c,c,c)=c$, so comparison with the constants $m,M$ proves $m\le U_j^n\le M$ for every time level. In particular the same <CFL condition> remains applicable. This is a mesh-independent maximum-principle <stability> bound.
A stronger <stability> statement is contraction in discrete $L^1$. Consider two solutions $U,V$ in the same invariant interval, on a periodic mesh or on the infinite lattice with summable differences. Let $Q=U\vee V$ be their componentwise maximum. <Monotonicity> gives $TQ\ge TU,TV$, hence
$$
\sum_j(TU_j-TV_j)^+\le\sum_j(TQ_j-TV_j)
=\sum_j(Q_j-V_j)=\sum_j(U_j-V_j)^+.
$$
The discrete <conservation law> for the difference gives the equality: on a finite periodic mesh the flux sum cancels exactly, and on the infinite lattice it follows by truncation and the fact that the flux is a <Lipschitz function> on $[m,M]$. Interchanging $U,V$ and adding proves
$$
\boxed{h\sum_j|U_j^{n+1}-V_j^{n+1}|\le h\sum_j|U_j^n-V_j^n|.}
$$
This is <L1 contraction of a monotone conservative scheme>. Since $T$ commutes with the cell shift, applying this contraction to $U$ and its shift gives $\sum_j|U_{j+1}^{n+1}-U_j^{n+1}|\le\sum_j|U_{j+1}^n-U_j^n|$. The scheme is therefore a <total variation diminishing scheme> as well.
The same order argument proves a <discrete entropy inequality>, explaining why this stable discretization selects <entropy solutions> rather than nonphysical expansion <shock waves>. For a constant $c\in[m,M]$, set
$$
Q_c(a,b)=F(a\vee c,b\vee c)-F(a\wedge c,b\wedge c).
$$
Because $T(U\vee c)\ge TU,c$ and $T(U\wedge c)\le TU,c$, their componentwise difference is at least $|TU-c|$. Subtracting their updates gives
$$
|U_j^{n+1}-c|\le|U_j^n-c|-r\bigl[Q_c(U_j^n,U_{j+1}^n)-Q_c(U_{j-1}^n,U_j^n)\bigr].
$$
For constants outside $[m,M]$, the entropy inequality is an equality by the invariant bound and the conservative update. At equal states $Q_c(u,u)=\operatorname{sgn}(u-c)[f(u)-f(c)]$, the <Kruzhkov entropy flux>. The flux is a <Lipschitz function> on the invariant interval: writing $L=\max|f'|$, the conservative update gives $h\sum_j|U_j^{n+1}-U_j^n|\le2kL\sum_j|U_{j+1}^n-U_j^n|$. This time-translation bound, the invariant bound and the <total variation> estimate give local space-time <compactness> for bounded-variation <initial conditions> as the mesh is refined; the conservative update passes to the weak <conservation law>, and the displayed inequality passes to its <Kruzhkov entropy inequality>. This identifies the limit as the <entropy solution>.
Finally the time-step qualification is essential. For the convex <Inviscid Burgers equation> flux $f(u)=u^2/2$, whose unique <sonic point of a scalar flux> is zero, linearize about a positive constant $U$. The perturbation update becomes $v_j^{n+1}=(1-rU)v_j^n+rUv_{j-1}^n$. At <Fourier frequency> $\pi$ its multiplier is $1-2rU$, with modulus greater than one if $rU>1$. <Wave packets> near this phase give <linear instability> even for summable Cauchy perturbations. Thus the assumptions on $f$ imply the <stability> results above under the <CFL condition>, not for arbitrary $k/h$. This is <CFL necessity for explicit Engquist-Osher stability>.
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