CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > Main CFD Forum

What's the mathematical meaning of characteristic decomposition of Euler equations

Register Blogs Members List Search Today's Posts Mark Forums Read

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   September 2, 2018, 23:46
Default What's the mathematical meaning of characteristic decomposition of Euler equations
  #1
Senior Member
 
Join Date: Oct 2017
Location: United States
Posts: 229
Blog Entries: 1
Rep Power: 8
TurbJet is on a distinguished road
Greetings,

When people apply nonreflecting BCs for compressible Euler equations, typically what we do is studying the characteristic decomposition of the compressible Euler equations, break the coefficient matrix into a diagonal matrix.

But what baffles me is that, what's the mathematical meaning of this decomposition? What do those variables (namely the eigenvalues and eigenvectors from the decomposition) mean mathematically? Why in this way can we impose BCs?

Thanks.
TurbJet is offline   Reply With Quote

Old   September 3, 2018, 04:16
Default
  #2
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,266
Rep Power: 67
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
Quote:
Originally Posted by TurbJet View Post
Greetings,

When people apply nonreflecting BCs for compressible Euler equations, typically what we do is studying the characteristic decomposition of the compressible Euler equations, break the coefficient matrix into a diagonal matrix.

But what baffles me is that, what's the mathematical meaning of this decomposition? What do those variables (namely the eigenvalues and eigenvectors from the decomposition) mean mathematically? Why in this way can we impose BCs?

Thanks.





From a mathematical point of view, by using the characteristic method you can reduce the PDE system to an ODE system with initial value problem.
In terms of linear algebra, the original matrix is pre and post multiplicated my matrices of eigenvectors to get a diagonal matrix where only eigenvalue appear. Of course, the resolved variables are not the original one but the transformed one. That is also at the origin of the Riemann invariant for omo-entropic flows. Such variables are called invariant because they remain constant along the characteristic curves, allowing for an exact mathematical solution. However, this can be done exactly only for particular 1D assumption.
FMDenaro is offline   Reply With Quote

Old   September 3, 2018, 11:11
Default
  #3
Senior Member
 
Join Date: Oct 2017
Location: United States
Posts: 229
Blog Entries: 1
Rep Power: 8
TurbJet is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
From a mathematical point of view, by using the characteristic method you can reduce the PDE system to an ODE system with initial value problem.
In terms of linear algebra, the original matrix is pre and post multiplicated my matrices of eigenvectors to get a diagonal matrix where only eigenvalue appear. Of course, the resolved variables are not the original one but the transformed one. That is also at the origin of the Riemann invariant for omo-entropic flows. Such variables are called invariant because they remain constant along the characteristic curves, allowing for an exact mathematical solution. However, this can be done exactly only for particular 1D assumption.
Thanks for the reply! And do you have any idea about the mathematical or physical meaning of left & right eigenvectors? This is what most people have used when they do characteristic decomposition.
TurbJet is offline   Reply With Quote

Old   September 3, 2018, 12:40
Default
  #4
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,266
Rep Power: 67
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
Quote:
Originally Posted by TurbJet View Post
Thanks for the reply! And do you have any idea about the mathematical or physical meaning of left & right eigenvectors? This is what most people have used when they do characteristic decomposition.

It is a linear algebra topic. Consider the matrix A and the following eigenvalues problems where the right and left eigenvectors appear




(A- lambda(k) I).rk = 0
lk^T .(A- lambda(k) I) = 0

so that you get a basis in R^N since lj^T.rk = 0.

After determining the eigenvectors, you can now build the matrix R =[r1,r2,...rN] and L =R^-1 such that



D = L.A.R

is the diagonal matrix having the eigenvalues as entries.
Therefore, the eigenvectors are used to build rotation matrices in such a way to express the diagonal form. Practically, you are rewriting the original matrix in terms of the principal axes




FMDenaro is offline   Reply With Quote

Old   September 3, 2018, 22:07
Default
  #5
Senior Member
 
Join Date: Oct 2017
Location: United States
Posts: 229
Blog Entries: 1
Rep Power: 8
TurbJet is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
It is a linear algebra topic. Consider the matrix A and the following eigenvalues problems where the right and left eigenvectors appear




(A- lambda(k) I).rk = 0
lk^T .(A- lambda(k) I) = 0

so that you get a basis in R^N since lj^T.rk = 0.

After determining the eigenvectors, you can now build the matrix R =[r1,r2,...rN] and L =R^-1 such that



D = L.A.R

is the diagonal matrix having the eigenvalues as entries.
Therefore, the eigenvectors are used to build rotation matrices in such a way to express the diagonal form. Practically, you are rewriting the original matrix in terms of the principal axes




So the sole purpose of using left/right eigenvectors is just to find out the principle axis of the matrix? Do the vectors themselves have some physical meaning?
TurbJet is offline   Reply With Quote

Old   September 4, 2018, 04:34
Default
  #6
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,266
Rep Power: 67
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
the eigenvectors are tangent to the integral curves in the the variable space, have a look to the book of Leveque
FMDenaro is offline   Reply With Quote

Old   September 4, 2018, 15:13
Default
  #7
Senior Member
 
Martin Hegedus
Join Date: Feb 2011
Posts: 500
Rep Power: 18
Martin Hegedus is on a distinguished road
The eigenvectors can be thought of as filters. The state variables are dotted with the eigenvector to get the variable of interest.

The physical variables of interest are entropy, vorticity, and the +/- acoustic waves.

At this time, I'm going to probably do some fudging. We always seem to use the term "eigenvector", however, I view the first three eigenvectors (the ones for entropy and vorticity which are multiplied by velocity) as an eigentensor. Not sure if eigentensor is an actual mathmatical term, but that's what it seems, at least to me, to be.
Martin Hegedus is offline   Reply With Quote

Reply

Tags
boundary condition, characteristics, compressible flow problem, euler equations

Thread Tools Search this Thread
Search this Thread:

Advanced Search
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
SU2 in Euler form coupled with BL equations francescofaggiano SU2 0 May 4, 2016 05:05
Normalization of eigenvectors of the Euler equations bubble45 Main CFD Forum 9 March 16, 2015 18:09
Euler Equations Boundary Conditions Christian87 Main CFD Forum 0 April 5, 2013 14:51
Delta form of Heat, Euler and NS equations RameshK Main CFD Forum 3 May 30, 2012 11:41
Euler equations? Jan Ramboer Main CFD Forum 2 August 19, 1999 02:58


All times are GMT -4. The time now is 09:28.