[cfdnewb123]

For the exact Riemann solver, it mainly involves calculating the p* and subsequently u*. Once p* is obtained, it is used to define the parameters in the other cells. However, how do I extend it to flux like format?

Because for approximate Riemann solver used in conjunction with Godunov method, the approximate Riemann solver are mostly used to obtain the flux at cell using values from left and right cell interfaces.

How do I modify the exact Riemann solver such that it gives me the flux at the cell? I suspect this will require iteration at every cells in order to compute p* at each cell.

I think it is obtained by
F=[rho*u*,rho*u*^2+p*,u*(E*+p*)] whereby u* can be obtained from p*. However, how to obtain rho*?

[naffrancois]

Hello,

Yes, once you get the full * solution your flux is obtained as you wrote. However, getting p* and u* is only the first step. You need to get rho* and to do this you need the relations across the rarefaction wave using Riemann invariants or across the shock using Rankine-Hugoniot jump relations. Then E* is obtained using the equation of state.

This is very well explained in Riemann Solvers and Numerical Methods for Fluid Dynamics by E. Toro.

[cfdnewb123]

Thanks for the reply. I try implementing exact riemann solver as flux in conjunction with godunov method and it seems to be at most as accurate as Roe solver. Is it possible to make the exact Riemann solver flux as accurate as just using exact Riemann solver without Godunov method (whereby p* is only obtained once)?

[Eifoehn4]

Originally Posted by cfdnewb123

Thanks for the reply. I try implementing exact riemann solver as flux in conjunction with godunov method and it seems to be at most as accurate as Roe solver. Is it possible to make the exact Riemann solver flux as accurate as just using exact Riemann solver without Godunov method (whereby p* is only obtained once)?

The exact flux should in most cases be more accurate than the Roe solver, since you compare

• the exact flux of the original problem with
• the exact flux of the linearized problem.

Curiously, even the exact Riemann flux violates some entropy constraints if used as flux function. Moreover, you won't get the same accuracy if you compare your CFD simulation to the exact Riemann solution, since

• the latter one is a microscopic solution (independent of the mesh), whereas
• the other one is a macroscopic solution (and mesh dependent).

Regards