Roe matrix for Finite volume scheme.

dokeun · December 5, 2012, 23:41

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, 12:52

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, 08:00

Quote:

Originally Posted by dokeun

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 11, 2012, 23:58

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.

Thank you.

ripperjack · December 12, 2012, 10:09

Quote:

Originally Posted by dokeun

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.

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

is the mth right eigenvector based on the Roe averaged matrix $\tilde{A}$ ,

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!

koderer · March 1, 2018, 07:12

You mentioned the Matrix T of the right eigenvectors and the book of Blazek.
There's a big question that bothers me for years:
Did somebody realize that in some components of the Matrix T in Blazek's book (formula A. 66, p. 426, 1st ed. 2001) the units are non consistent?
I mean in the first three columns, rows 2 to 4, there are matrix-entries like
$T_{31}=n_x v+n_z \rho$
How can a quantity of units [velocity] be added to a quantity of units [density]?
The same question applies to the book of Hirsch (vol. 2, formula (16.5.22), p. 181), which shows basically the same T-matrix (it's called P there and scaled by factor $1\slash\sqrt{2}$ ). By the way, Hirsch is the only source where substeps for the derivation are given in addition to the resulting eigenvector matrices. Does anybody know another place where the derivation of the eigenvectors is explained step by step?

sbaffini · March 1, 2018, 11:00

The very point on eigenvectors is that they are not uniquely determined, so different results can be obtained. The one I like more is:

Balasubramanian, Chen: Preconditioned Algorithms with a General Equation of State for Rotating Machinery Flows, AIAA 2010-4862.

EDIT: I don't have confidence with those formulations showing inconsistencies, so I cannot give help on that.

December 5, 2012, 23:41	Roe matrix for Finite volume scheme.	#1
dokeun Member Dokeun, Hwang Join Date: Apr 2010 Location: Korea, Republic of Posts: 98 Rep Power: 16	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.

March 1, 2018, 07:12	inconsistent units in eigenvector matrix by Blazek or Hirsch	#6
koderer New Member Join Date: Jan 2012 Posts: 9 Rep Power: 14	You mentioned the Matrix T of the right eigenvectors and the book of Blazek. There's a big question that bothers me for years: Did somebody realize that in some components of the Matrix T in Blazek's book (formula A. 66, p. 426, 1st ed. 2001) the units are non consistent? I mean in the first three columns, rows 2 to 4, there are matrix-entries like $T_{31}=n_x v+n_z \rho$ How can a quantity of units [velocity] be added to a quantity of units [density]? The same question applies to the book of Hirsch (vol. 2, formula (16.5.22), p. 181), which shows basically the same T-matrix (it's called P there and scaled by factor $1\slash\sqrt{2}$ ). By the way, Hirsch is the only source where substeps for the derivation are given in addition to the resulting eigenvector matrices. Does anybody know another place where the derivation of the eigenvectors is explained step by step? Last edited by koderer; March 2, 2018 at 17:52.

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December 6, 2012, 12:52		#2
Vasiliy New Member Vasiliy Join Date: Feb 2011 Posts: 9 Rep Power: 15	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.

December 11, 2012, 23:58		#4
dokeun Member Dokeun, Hwang Join Date: Apr 2010 Location: Korea, Republic of Posts: 98 Rep Power: 16	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. Thank you.

March 1, 2018, 11:00		#7
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,152 Blog Entries: 29 Rep Power: 39	The very point on eigenvectors is that they are not uniquely determined, so different results can be obtained. The one I like more is: Balasubramanian, Chen: Preconditioned Algorithms with a General Equation of State for Rotating Machinery Flows, AIAA 2010-4862. EDIT: I don't have confidence with those formulations showing inconsistencies, so I cannot give help on that.