Numerical flux ($F(a,b)$)

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.