Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 57 / 9 / Solution

Let $s_{\ast }$ be the unique sonic point, $f^{\prime }(s_{\ast })=0$. For a differentiable convex function, $f^{\prime }$ is nondecreasing, negative below $s_{\ast }$ and positive above it. Split the flux into the parts propagating right and left:Thus $f(u)=f(s_{\ast })+f^{+}(u)+f^{-}(u)$; explicitly, $f^{+}(u)=f(u)-f(s_{\ast })$ for $u>s_{\ast }$ and zero below, while $f^{-}(u)=f(u)-f(s_{\ast })$ for $u<s_{\ast }$ and zero above. The Engquist-Osher flux takes the right-going contribution from the left state and the left-going contribution from the right state:It is consistent, $F(u,u)=f(u)$. For a transonic rarefaction wave with $a\le s_{\ast }\le b$ it gives $f(s_{\ast })$; reversed transonic states give $f(a)+f(b)-f(s_{\ast })$. This is sonic-point splitting of a convex Engquist-Osher flux.

On cells of width $h$ let $U_j^n$ approximate the cell average and $\lambda =\Delta t/h$. The explicit Engquist-Osher method is the finite volume methodThe shared interface flux makes it conservative: sums telescope for periodic data, and the same applies on the infinite grid when boundary contributions vanish. To prove stability, choose an interval $[m,M]$ containing the initial states and impose the CFL conditionThe partial derivatives of the update areTherefore the update is a monotone conservative scheme. Because $H(q,q,q)=q$, componentwise comparison with the constant states $m,M$ proves $m\le U_j^{n+1}\le M$. Induction gives a mesh-independent maximum bound for every time level, and ensures the same CFL speed bound remains valid.

There is also a stronger L1 contraction of a monotone conservative scheme. Let $T$ denote one whole-grid update, and let $W=\min (U,V)$ and $Z=\max (U,V)$ componentwise. Monotonicity gives $TW\le TU,TV\le TZ$. Hence $|TU-TV|\le TZ-TW$ componentwise, and conservation yieldsThis applies to periodic grids or summable perturbations with vanishing end fluxes. Apply the same result to $U$ and its one-cell translate; translation commutes with $T$, soThus the method is a total variation diminishing scheme. Under the stated CFL condition it preserves the initial range, contracts differences in discrete $L^1$, and cannot increase total variation. These bounds give a precise nonlinear stability result, including across the sonic point.

Convexity alone does not remove the time-step restriction omitted in the question. For the Inviscid Burgers equation $f(u)=u^2/2$, linearize about a positive constant $U$. The update becomes $v_j^{n+1}=(1-\lambda U)v_j^n+\lambda U{v}_{j-1}^n$. Its alternating Fourier mode has multiplier $1-2\lambda U$, whose modulus exceeds one if $\lambda U>1$. Hence unrestricted explicit time steps are unstable even for a convex flux with one sonic point. This is CFL necessity for explicit Engquist-Osher stability.