Some numerical schemes, in particular finite volume methods, are attributed as conservative. But what does that mean exactly? When I click on the Wikipedia page, it redirects to physical conservation laws. But I want to learn something about the numerical meaning of conservative.
For example, take the Poisson equation and the heat equation:
$\nabla u = f$
$\nabla v = \frac{\partial v}{\partial t}$
Now a numerical scheme to solve these equations will give us two grid functions $u^h, v^h$. In what aspect will a solution achieved with a conservative scheme differ from a solution achieved with a scheme that is not necessarily conservative?
Best Answer
I think conservative is used in different ways. You may think indeed of conservation of global quantities, but I think that in most theoretical works on finite volumes, conservative means "with consistent and conservative numerical fluxes".
In the Lax-Wendroff theorem for hyperbolic conservation laws, which is often stated as:
"if a conservative numerical scheme for a hyperbolic system of conservation laws converges, then it converges towards a weak solution," (wikipedia)
here, conservative means that the numerical flux from cell K to L is the opposite as the numerical flux from cell L to K. Of course this implies global conservation.
In fact, the assumptions of the Lax-Wendroff theorem also requires the flux to be consistent, in the sense that if the equation reads $\partial_t + \partial_x(f(u)) = 0$, and the flux is say a two point flux $g(u_K, u_L)$, then it must satisfy $g(u,u)= f(u)$ (it is a one dimensional multipoint flux in the original Lax Wendroff paper).
Conservative for elliptic problems has the same meaning, i.e. conservative flux. The finite volume for the Poisson equation of the heat equation is not consistent in the classical finite difference sense on non uniform grids (so the Lax-Ritchmyer theorem does not apply), but it is convergent thanks to the conservativity and the consistency of the fluxes. More on this in our chapter of Handbook of Numerical Analysis, available on the web. https://old.i2m.univ-amu.fr/~herbin/PUBLI/bookevol.pdf