Finite volume method

A finite volume method evolves cell integrals or cell averages by differences of shared interface numerical fluxes. For a scalar conservation law, $U_j^{n+1}=U_j^n-(k/d)(F_{j+1/2}-F_{j-1/2})$. Sharing the same flux between neighboring cells makes conservation follow by telescoping.

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.

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 $\frac{1}{2}\max (a,0{)}^2+\frac{1}{2}\min (b,0{)}^2$.

A numerical flux approximates the transported quantity through a cell interface using nearby states. For a two-state flux $F(a,b)$ approximating a scalar conservation law with flux $f$, consistency requires $F(u,u)=f(u)$. A monotone conservative scheme uses a flux nondecreasing in its left state and nonincreasing in its right state, with a time step making the complete update monotone.