How does positivity preservation fit into the implication chain from monotone to monotonicity preserving?

Question

I know from "Numerical Methods for Conservation Laws" by Randall J. LeVeque that there is an implication chain of properties of methods for conservation laws:

monotone $\Rightarrow$ $L^1$-contractive $\Rightarrow$ TVD $\Rightarrow$ monotonicity preserving

For the conservation of mass, positivity preservation is most important, but how does it fit into this chain? Does monotonicity preservation imply positivity preservation or vice-versa? A lot of people seem to use both terms as equivalent, but I'm not so sure that's true...

Answer 1

Of course, all methods in this area conserve mass. For scalar conservation laws, conservation of mass plus $L^1$ contractivity implies positivity. But for systems of conservation laws, it does not. In practice, schemes have been designed to preserve positivity for Euler and shallow water flows by many means, most of which involve special Riemann solvers.