Roe matrix for Finite volume scheme.
From J.Blazek's, intercell flux for roe is expressed as a production of roe matrix and difference of conservation variables between left & right
So, I'm trying to derive eigenvalues, eigenvectors, wave strength from roe matrix written for 2D Euler Finite volume scheme. But I confused whether I'm doing right.
Many references explain the conservation lows and roe flux in serperated coordinates. From Toro, there is a method to express these flux terms as one augmented 1D flux.
Where can I find a whole process to derive roe matrix for the augmented 1D flux!?
Any kind of advice will be a help. Thank you in advance.
Try to find Weiss and Smith paper "Preconditioning Applied to Variable and Constant Density Time-Accurate Flows on Unstructured Meshes".
May by it can be useful for you.
To find more details about Roe scheme, the chapter 11 from "Riemann Solvers and Numerical Methods for Fluid Dynamics A Practical Introduction" by Toro, E. F. will be very helpful!
Dear Vasiliy, ripperjack
Thank you for your advices. I just read the paper and the took.
But I think they didn't resolve my concern. Maybe I didn't describe the exact point I stuck.
So, I'd like to change my question.
The discretized integral form of conservation laws(2d euler) for unstructured scheme is
In order to find inter cell flux by roe shceme, I needed to fined and which satisfy following relations,
Paramether vector ,
from these I can find roe matrix .
But, the problem is that it's too complicate to find from flux .
I'd like to know that obtaining from above relation is straightforward despite of its complexity.
For the 1st row of , I found but I couldn't for the other rows. It's too complicated.
If the fluxs are splitted x, y direction(structured scheme) I have no problem with it.
I'd like to know if I'm going a right way. :confused:
It seems that you are trying to derive Roe matrix , and calculate the flux at the cell surfaces.
I did not know the details of derivation of Roe matrix , I just used it.
And is exact the same as the original convective Jacobian A in 2D Euler equation, except that the flow variables are replaced with the Roe averaged variables (see J.Blazek's book, p108).
And the flux at the cell surface can be calculate by:
The is the mth right eigenvector based on the Roe averaged matrix ,
The is the mth eigenvalue also based on the
The wave strength is defined by:
so the wave strength can be calculated by:
and is the left eigenvector based on the
The left and right eigenvector can be find in J.Blazek's book A.11
In addition, if you just want to use Roe scheme, the Eq. 4.91-4.95 in J.Blazek's book can be used directly. I used that equations and they worked good.
That is all I can do, hope that would help!:)
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