Conservation properties of RBF-FD

[kalosu]

Hi there,

I have been using your nice RBF-FD package to solve some linear PDE systems. Thanks a lot for making this available to everyone.

Recently I came across the finite volume method, from which I understood that one of the key advantages of this other method is that it inherently guarantees conservation properties of the system (i.e, for instance mass, energy density, etc). What about the case of RBF-FD? Do you know if these properties are also hold to RBF-FD?

From the formulation point of view I see that both of them are different (even though at the end you approximate derivatives in both cases, in FVM you approximate the derivative of fluxes if I am not mistaken)

Maybe you have some comments about this?

[treverhines]

I don't think there is anything about the rbf-fd formulation that would inherently conserve those quantities. I think the relevant difference is that the rbf-fd method just has nodes and no cells, meaning the rbf-fd method doesn't even provide a clear way to integrate over the domain to check if those quantities are conserved.

[kalosu]

I see, thanks for the feedback.

What would then this mean if I am trying to solve a PDE for which conservation of mass is important?

I'm trying to implement an optimal mass transport algorithm which on the original paper they used FVM to solve the needed linear systems.

In the past, I implemented an older version of this algorithm using RBF-FD but being honest, I never paid attention to this until now.

So that basically means: I dont even know if this has an impact on the accuracy of the solutions.

Probably I would have to compare both approaches? (RBF-FD vs FVM) And try to define some way to evaluate conservation properties for the obtained solutions right?