7
$\begingroup$

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...

$\endgroup$
1
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.