I thought finite volume solves the integral version of the conservation equations. yet in papers on upwind/symmetric TVD schemes it solves the normal conservation form using metrics? Are these metrics defined at the cell faces or what?

Shuo

For example in a paper by Daru & Tenuad (Computers & Fluids 30:89-113; 2001), The discrete equation reads:

dw/dt = -1/delta_x*(f_i+0.5 - f_i-0.5 + fv_i+0.5 - f_i-0.5) - 1/delta_y*(g_j+0.5 - g_j-0.5 + gv_j+0.5 - gv_j-0.5)

w is the conservative variables, f and g are the inviscid fluxed, fv and gv are the viscous fluxes.

isn't this just a finite difference discretisation?

Shuo

I guess that you are confused between cell-centered schemes and cell-vertex schemes. In compressible simulations, cell-centered schemes are common. As you posted, it looks like a cell-centered scheme. See, pp.85-94 in "computational fluid dynamics: principles and applications" written by J. Blazek

Oh, dear now i need to do a bit of editing of my code. For cell centered schemes is the boundary condition applied to the u_(i+0.5) instead of u_i?

Shuo

Physically, yes. However, it may not make significant impact on the boundary effect. In my cases, I just impose B.C to u(i). In other way, you can use extrapolated values.

Impose it on u_(i+0.5), i.e., on the fluxes. THis is more consistent with the finite volume method.

You are confusing terminology a little.

When you write a discrete flux approximation in this way it is said to be in "conservation form". This is hold over from the early days (1970s to mid 1980s) when no-one ever thought much about unstructured meshes for CFD.

So, written this way it is a finite volume scheme but is really only valid for a uniform, orthogonal, Cartesian mesh.

You would write it much different for a non-uniform unstructured mesh.