= Solution
The <Engquist-Osher method> is a conservative <finite volume method> built by separating positive and negative characteristic speeds. Take $f\in C^1$, a uniform cell width $d$, a time step $k$, and cell averages $U_j^n$. Write $\lambda=k/d$ and define the <numerical flux>
$$
F(a,b)=f(0)+\int_0^a\max(f'(s),0)\,ds
+\int_0^b\min(f'(s),0)\,ds.
$$
The <Engquist-Osher flux> uses the left state for right-going transport and the right state for left-going transport. Its conservative update is
$$
U_j^{n+1}=U_j^n-\lambda\bigl(F(U_j^n,U_{j+1}^n)-F(U_{j-1}^n,U_j^n)\bigr).
$$
It is consistent because $F(u,u)=f(u)$. The integrals have their ordinary oriented meaning even for negative states. For the <Inviscid Burgers equation>, $f(u)=u^2/2$ gives the useful example
$$
F(a,b)=\frac12\max(a,0)^2+\frac12\min(b,0)^2.
$$
In a region where all characteristic speeds have one sign, the flux reduces to the corresponding upwind flux. In smooth regions, this basic piecewise-constant, forward-time version is first order in space and time; its main virtue is robust nonlinear <stability> across shocks.
Here is a precise <stability> proof. Assume the data lie in a bounded interval $J=[m,M]$ and choose
$$
\boxed{\lambda\sup_{s\in J}|f'(s)|\leq1.}
$$
Work on a periodic grid, or on the infinite grid with summable differences; on a finite interval the inflow boundary data and boundary fluxes must be treated monotonically as well. Define the three-point update map
$$
H(a,b,c)=b-\lambda\bigl(F(b,c)-F(a,b)\bigr).
$$
Writing $f'_+=\max(f',0)$ and $f'_- =\min(f',0)$, its three partial derivatives are
$$
H_a=\lambda f'_+(a)\geq0,\qquad
H_b=1-\lambda|f'(b)|\geq0,\qquad
H_c=-\lambda f'_-(c)\geq0.
$$
Thus it is a <monotone conservative scheme>. Since $H(s,s,s)=s$, comparison with the constant states proves
$$
m\leq U_j^n\leq M\quad\hbox{for all }j,n.
$$
The interval is invariant, so the same <Courant–Friedrichs–Lewy condition> remains valid at every step. This is an $L^\infty$ bound relative to constant states, not a claim that arbitrary pairs of solutions are contractive in $L^\infty$.
For a stronger perturbation estimate, let $\mathcal T$ denote the global update and use componentwise maxima. Monotonicity gives $\mathcal T(U\vee V)\geq\mathcal TU,\mathcal TV$. Therefore
$$
(\mathcal TU-\mathcal TV)_+\leq\mathcal T(U\vee V)-\mathcal TV.
$$
Sum over the grid. The shared interface flux cancels telescopically, so conservation gives
$$
\sum_j(\mathcal TU_j-\mathcal TV_j)_+
\leq\sum_j(U_j-V_j)_+.
$$
Interchanging $U,V$ and adding yields the <L1 contraction of a monotone conservative scheme>:
$$
\boxed{d\sum_j|U_j^n-V_j^n|\leq d\sum_j|U_j^0-V_j^0|.}
$$
This is a mesh-independent nonlinear <stability> estimate. On an infinite grid the telescoping argument is justified by summability of the differences and the bounded characteristic-speed range, or by truncation followed by a limit. For unequal prescribed boundary data, additional boundary flux terms enter the estimate.
The method is also a <total variation diminishing scheme>. Let $\mathcal S$ shift a grid sequence by one cell. Translation invariance gives $\mathcal T\mathcal S=\mathcal S\mathcal T$. Applying the preceding $L^1$ contraction to $U$ and $\mathcal SU$ shows
$$
\sum_j|U_{j+1}^{n+1}-U_j^{n+1}|
\leq\sum_j|U_{j+1}^{n}-U_j^{n}|.
$$
Hence initially finite discrete <total variation> cannot grow.
Finally, monotonicity supplies a <discrete entropy inequality>, explaining why the <stability> is compatible with the physically admissible <entropy solution>. For a constant $c\in J$, define
$$
Q(a,b;c)=F(a\vee c,b\vee c)-F(a\wedge c,b\wedge c).
$$
Comparison of $H$ with the clipped triples above and below $c$ gives
$$
|U_j^{n+1}-c|\leq|U_j^n-c|
-\lambda\bigl(Q(U_j^n,U_{j+1}^n;c)-Q(U_{j-1}^n,U_j^n;c)\bigr).
$$
Indeed $H(U\vee c)\geq\max(H(U),c)$ and $H(U\wedge c)\leq\min(H(U),c)$, and subtracting these two update formulas yields exactly the right side. Also $Q(u,u;c)=\operatorname{sgn}(u-c)(f(u)-f(c))$, the <Kruzhkov entropy flux>. Under the displayed CFL condition the method therefore has an invariant state range, $L^1$ contraction, decreasing discrete <total variation> and entropy dissipation. These estimates remain meaningful at discontinuities, where linearized <Fourier analysis> alone cannot establish the corresponding nonlinear <stability>.
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