Finite Difference on unstructured grids ?
 

Hello,

I've encountered an interesting passage in the book of Ferziger and Peric about the possibility of using "finite difference" discretizations on unstructured grids. I'm curious if someone has tried this approach.

At a first look this could be done by fitting a polynomial on a set of points surrounding the point of interest (for a triangular grid these points can be the neighbour points).

Do you think this approach can be applied for transonic flows with shocks ?

Do

I haven't done this yet, and I can't remember ever seeing people at conferences and such doing unstructured FD.... I guess it would lose most of its simplicity if you would have to fiddle around with building polynomials in arbitrary space.... from my opinion, if you have to do that, why not go the rest of the way and do FE?

I think you are right about losing some of the simplicity of the formulation if you apply FD on unstructured grids.

I wonder if you could arbitrarily increase the order of precision of a scheme on unstructured grids the way you can on structured grids ... I suppose a better way to achieve higher precision will be to use DG FEM schemes or spectral methods.

Do

I guess you can indeed increase the order in an unstructured setting with FD - I guess it just becomes too messy and convoluted to be effective....
in my part of the community, people only use FD on structured grids with medium order (6, 8, sth like that) to do research into basic turbulence and transition. As soon as the geometry gets a little bit more elaborate, they turn to DGFEM or DGSEM.

Cheers,
newbie

finite difference use Taylor expnsion series to find out difference scheme. it needs orthonal grid. so "unstuructured mesh" not usually orthogonal . so not valid for implementation fd on "unstructured....

Well, not fully true. You can construct an orthogonal basis in space, and then interpolate your unstructured nodes.... FEM does that for example. So it's possible, just not in an easy or efficient way for FD- as far as I know!

cheers!

You may search by 'wlsqr scheme' and for certain reason they make it finite volume, probably to account for conservation, but finite difference also possible. I also saw an application of the method to transonic flows on recent conference, they used no any mesh, just distribution of points. And i also saw a paper using radial basis functions instead of polynomials. I may refind those links if you are interested.

@carambula

Thanks,

I think you are talking about meshless (or mesh free) methods which are a bit different from what is suggested in Ferziger and Peric's book. (Actually they are also talking about mesh free methods, but this is a separate paragraph and a different idea than applying FD on unstructured grids.)

Do

@DoHander

maybe it's 'mimetic finite difference'?

It is very much feasible
 

It can be done, and it is efficient indeed. It's simple and straight forward. In FE stiffness matrix is built, but FD you build the differential terms directly.
I use it this way too.