= CFL necessity for explicit Engquist-Osher stability
{c}
{title2=$\frac{k}{h}\sup|f'|\le1$}
<Monotonicity> of the explicit update requires $(k/h)\sup_{[m,M]}|f'|\le1$ 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>: $v_j^{n+1}=(1-rU)v_j^n+rUv_{j-1}^n$, 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>.
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