 CFL necessity for explicit Engquist-Osher stability

Monotonicity of the explicit update requires $(k/h){\sup }_{[m,M]}|f^{\prime }|\le 1$ on its invariant state interval. Convexity and a unique sonic point of a scalar flux do not remove this time-step restriction. For $f(u)=u^2/2$, linearization about a positive constant $U$ gives an upwind finite difference scheme for advection equation: $v_j^{n+1}=(1-rU)v_j^n+rU{v}_{j-1}^n$, where $r=k/h$. At Fourier frequency $\pi$ the multiplier is $1-2rU$, whose modulus exceeds one when $rU>1$. Thus a CFL condition is essential to the claimed nonlinear stability.