Monotone conservative scheme

A scalar finite-volume update is monotone if increasing any input state cannot decrease any output state. A consistent conservative update preserves constants and has an invariant interval by comparison. On a periodic grid, or an infinite grid with summable perturbations, conservation and monotonicity imply L1 contraction of a monotone conservative scheme. Translation invariance then implies the total variation diminishing scheme property.

A translation-invariant conservative monotone update decreases discrete total variation. If $S$ is the one-cell shift, $TS=ST$, and L1 contraction of a monotone conservative scheme applied to $U,SU$ gives ${\sum }_j|(TU{)}_{j+1}-(TU{)}_j|\le {\sum }_j|U_{j+1}-U_j|$. This prevents creation of new total variation, though it does not make every discontinuity sharply resolved.

Let $T$ be a monotone conservative scalar update on a periodic grid. Since $T(U\vee V)\ge TU,TV$, one has $(TU-TV{)}_{+}\le T(U\vee V)-TV$. Conservation makes the sum of the right side equal to $\sum (U-V{)}_{+}$. Reverse $U,V$ and add to obtain $\Vert TU-TV{\Vert }_1\le \Vert U-V{\Vert }_1$. On an infinite grid the same proof holds for summable differences with a Lipschitz flux on the state range.

The Engquist-Osher method is the finite volume method using the Engquist-Osher flux. Under $(k/d)\sup |f\prime |\le 1$ on the invariant state interval, its update is monotone. It preserves the state range, is contractive for summable perturbations in $L^1$, and decreases discrete total variation.

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.

The Engquist-Osher numerical flux is $F(a,b)=f(0)+{\int }_0^a\max (f\prime (s),0)ds+{\int }_0^b\min (f\prime (s),0)ds$. It sends positive-speed contributions from the left state and negative-speed contributions from the right state. Its consistency $F(u,u)=f(u)$ and opposite monotonicities in the two arguments yield a monotone conservative scheme under the appropriate Courant–Friedrichs–Lewy condition. For the Inviscid Burgers equation, it is $\frac{1}{2}\max (a,0{)}^2+\frac{1}{2}\min (b,0{)}^2$.

For a flux that is a differentiable convex function with unique minimum at $s_{\ast }$, define $f^{+}(u)=f(u)-f(s_{\ast })$ for $u>s_{\ast }$ and zero otherwise, and $f^{-}(u)=f(u)-f(s_{\ast })$ for $u<s_{\ast }$ and zero otherwise. The Engquist-Osher flux is $F(a,b)=f(s_{\ast })+f^{+}(a)+f^{-}(b)$. Thus transsonic rarefaction wave states $a\le s_{\ast }\le b$ give $F=f(s_{\ast })$, while reversed transsonic states give $F=f(a)+f(b)-f(s_{\ast })$. This follows by integrating the positive and negative parts of $f^{\prime }$.