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dokeun December 6, 2012 00:41

Roe matrix for Finite volume scheme.
 
Hello

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.

Vasiliy December 6, 2012 13:52

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.

ripperjack December 11, 2012 09:00

Quote:

Originally Posted by dokeun (Post 396041)
Hello

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.



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!

dokeun December 12, 2012 00:58

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

U_{t} = - \frac{1}{\Omega_I} \sum^{N}_{m=1} \vec{F_c} \Delta S_m.

In order to find inter cell flux by roe shceme, I needed to fined \tilde{B}(\tilde{Q}) and \tilde{C}(\tilde{Q}) which satisfy following relations,

\Delta U = \tilde{B} \Delta \tilde{Q}, \Delta F_{c} = \tilde{C} \Delta \tilde{Q},

Paramether vector \tilde{Q} = \frac{1}{2}\left( \sqrt{\rho_{R}}+\sqrt{\rho_{L}} ~~~ \sqrt{\rho_{R}}u_{R}+\sqrt{\rho_{L}}u_{L} ~~~ \sqrt{\rho_{R}}v_{R}+\sqrt{\rho_{L}}v_{L} ~~~ \sqrt{\rho_{R}}H_{R}+\sqrt{\rho_{L}}H_{L} \right)^T = \left( \tilde{q}_1 ~~~ \tilde{q}_2 ~~~ \tilde{q}_3 ~~~ \tilde{q}_4 \right)^T,

And \Delta \tilde{Q} = \left( \sqrt{\rho_{R}}-\sqrt{\rho_{L}} ~~~ \sqrt{\rho_{R}}u_{R}-\sqrt{\rho_{L}}u_{L} ~~~ \sqrt{\rho_{R}}v_{R}-\sqrt{\rho_{L}}v_{L} ~~~ \sqrt{\rho_{R}}H_{R}-\sqrt{\rho_{L}}H_{L} \right)^T

from these I can find roe matrix \tilde{A} = \tilde{C} \tilde{B}^{-1}.

But, the problem is that it's too complicate to find \tilde{C} from flux \vec{F}_c.

Here, \vec{F}_c = \left( \rho V ~~~ \rho u V+n_{x}P ~~~ \rho v V+n_{y}P ~~~ \rho H V \right)^{T}.

I'd like to know that obtaining \tilde{C} from above relation is straightforward despite of its complexity.

For the 1st row of \tilde{C}, I found \left(n_{x}\tilde{q}_{2}+n_{y}\tilde{q}_{3} ~~~ n_{x}\tilde{q}_{1} ~~~ n_{y}\tilde{q}_{1} ~~~ 0 \right) 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:

Thank you.

ripperjack December 12, 2012 11:09

Quote:

Originally Posted by dokeun (Post 397055)
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

U_{t} = - \frac{1}{\Omega_I} \sum^{N}_{m=1} \vec{F_c} \Delta S_m.

In order to find inter cell flux by roe shceme, I needed to fined \tilde{B}(\tilde{Q}) and \tilde{C}(\tilde{Q}) which satisfy following relations,

\Delta U = \tilde{B} \Delta \tilde{Q}, \Delta F_{c} = \tilde{C} \Delta \tilde{Q},

Paramether vector \tilde{Q} = \frac{1}{2}\left( \sqrt{\rho_{R}}+\sqrt{\rho_{L}} ~~~ \sqrt{\rho_{R}}u_{R}+\sqrt{\rho_{L}}u_{L} ~~~ \sqrt{\rho_{R}}v_{R}+\sqrt{\rho_{L}}v_{L} ~~~ \sqrt{\rho_{R}}H_{R}+\sqrt{\rho_{L}}H_{L} \right)^T = \left( \tilde{q}_1 ~~~ \tilde{q}_2 ~~~ \tilde{q}_3 ~~~ \tilde{q}_4 \right)^T,

And \Delta \tilde{Q} = \left( \sqrt{\rho_{R}}-\sqrt{\rho_{L}} ~~~ \sqrt{\rho_{R}}u_{R}-\sqrt{\rho_{L}}u_{L} ~~~ \sqrt{\rho_{R}}v_{R}-\sqrt{\rho_{L}}v_{L} ~~~ \sqrt{\rho_{R}}H_{R}-\sqrt{\rho_{L}}H_{L} \right)^T

from these I can find roe matrix \tilde{A} = \tilde{C} \tilde{B}^{-1}.

But, the problem is that it's too complicate to find \tilde{C} from flux \vec{F}_c.

Here, \vec{F}_c = \left( \rho V ~~~ \rho u V+n_{x}P ~~~ \rho v V+n_{y}P ~~~ \rho H V \right)^{T}.

I'd like to know that obtaining \tilde{C} from above relation is straightforward despite of its complexity.

For the 1st row of \tilde{C}, I found \left(n_{x}\tilde{q}_{2}+n_{y}\tilde{q}_{3} ~~~ n_{x}\tilde{q}_{1} ~~~ n_{y}\tilde{q}_{1} ~~~ 0 \right) 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:

Thank you.

Dear Dokeun,

It seems that you are trying to derive Roe matrix \tilde{A}, and calculate the flux at the cell surfaces.
I did not know the details of derivation of Roe matrix \tilde{A}, I just used it.
And \tilde{A} 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:
F_{i+1/2}=0.5*(F(U_L)+F(U_R))-0.5*\sum^{N}_{m=1} {\alpha_m} \left| {\lambda_m}\right| T_m
The T_m is the mth right eigenvector based on the Roe averaged matrix \tilde{A}, T_m=(r_1,r_2,r_3,r_4)^T
The {\lambda_m} is the mth eigenvalue also based on the \tilde{A}
The wave strength {\alpha_m} is defined by:
\Delta \vec{U}=U_R-U_L= \sum^{N}_{m=1} {\alpha_m} T_m=\vec{T}\vec{\alpha}
so the wave strength {\alpha_m} can be calculated by:
\vec{\alpha}=\vec{T}^{-1}\Delta \vec{U}=({\alpha_1},{\alpha_2},{\alpha_3},{\alpha_4})^T
and \vec{T}^{-1} is the left eigenvector based on the \tilde{A}
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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