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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?

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  • $\begingroup$ This probably means that the scheme preserves the energy. The meaning of "energy" changes from equation to equation, and if the term comes from a mathematician, there is no guarantee that it refers to a true physical quantity. In the case of the Laplace equation $\Delta u=0$, "energy" surely refers to the integral $\int |\nabla u|^2\, dx$. For the Poisson equation you need to add a term dependent on $f$. And for the heat equation it might mean either $\int v^2\, dx$ or $\int |\nabla v|^2\, dx$. $\endgroup$ – Giuseppe Negro Nov 15 '17 at 10:08
  • $\begingroup$ How do you know that in the Laplace equation "energy" refers to that particular integral? Why the square and why the gradient? What does energy conservation mean in the Laplace equation? There is no evolution over time. Conservation implies that a quantity stays the same over different instances. However I only have one instance (the solution $u^h$) $\endgroup$ – don-joe Nov 15 '17 at 10:10
  • $\begingroup$ That's only a jargon, and it is context dependent. I listed some possibilities, but there's no guarantee that they are the only possible. The answer by user7530 is good and I agree with it. $\endgroup$ – Giuseppe Negro Nov 15 '17 at 10:14
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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

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I think you're overthinking the issue: a method is conservative if it yields a solution that obeys physical conservation laws.

One common setting is Hamiltonian systems, where one can ask for a numerical method for time integration that produces a discrete trajectory that obeys conservation of energy, momentum, and other first integrals. But again, the meaning of "conservative" is problem- and context-specific.

I don't know what "conservative" would mean for elliptic problems, but along the same lines, you can ask that the numerical method preserves important structure and invariants present in the exact solution; for instance for the Laplace equation you might want to guarantee that the numerical method always produces a solution that satisfies a maximum principle.

The common thread in all of these cases is that any reasonable numerical method will be convergent: the numerical solution will converge to the true solution as you refine the discretization (take smaller time steps, or use a finer spatial grid). Similarly if you have a conservation law, like conservation of energy, that law will be satisfied in the limit of refinement of the discretization. A conservative method will satisfy laws exactly at any level of refinement.

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  • $\begingroup$ So would you say, calling a scheme "conservative" for an elliptic problem or a steady state problem does not make sense? What does it mean that finite volume methods are "naturally" conservative? How do I know which physical conservation law they obey? $\endgroup$ – don-joe Nov 15 '17 at 10:16
  • $\begingroup$ @don-joe if I had to take a wild guess, I'd say the source was referring to conservation of volume. But you'd have to give a link to the context for me to know for sure. $\endgroup$ – user7530 Nov 15 '17 at 10:20

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