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Normalization of eigenvectors of the Euler equations 

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March 16, 2015, 11:34 
Normalization of eigenvectors of the Euler equations

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Tali Neuman
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Hello
I would like some help regarding the computation of eigenvalues for the implementation of Roe’s scheme for gas dynamics. I don’t know what is the role to normalize eigenvectors (left and right) so they will be suitable to Roe’s scheme. Thanks Tali Neuman p.s: I succeeded to: 1. Compute eigenvalues from the jacobian matrix of the non conservative form of the Euler equations. 2. Compute the left/right eigenvectors for the jacobian matrix of the non conservative form of the Euler equations. 3. Transform those eigenvectors to be suitable to the jacobian matrix of the conservative form of the Euler equations. 

March 16, 2015, 14:39 

#2  
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Filippo Maria Denaro
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Quote:
you can find the answer in the book of LeVeque, remeber you have one degree of freedom in determining the components of the eigenvector 

March 16, 2015, 16:26 
degree of freedom in determining the components of the eigenvector

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Tali Neuman
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Hello FMDenaro
Thanks for replying. Can you please point on a specific place in the book where there is a reference to this issue? Thanks a lot Tali Neuman 

March 16, 2015, 16:49 

#4  
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Filippo Maria Denaro
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have a look through the book, several parts can be useful for you, in particular tha anlysis of Euler equations 

March 16, 2015, 17:28 
Choice of the number for the degree of freedom component

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Tali Neuman
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Can I choose any number for the degree of freedom component of each of the right / left eigenvectors for the Roe’s scheme to work?


March 16, 2015, 17:32 

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Filippo Maria Denaro
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March 16, 2015, 17:44 
Choosing values for degree of freedom component

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Tali Neuman
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Thanks for replying
I would like to know if I understood well your answer: For the jacobian matrix of the 1D Euler equations, there are three eigenvectors. In each of them, there is one component which can be determined freely (=degree of freedom). After choosing freely those components for the right / left eigenvectors, can I use them in the Roe’s scheme? Thanks a lot 

March 16, 2015, 17:48 

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Filippo Maria Denaro
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remember that the expressions of the eigenvalues depends on the type of variables you are using [rho,u,p], [rho,u,s]. In the book you will find several numerical methods 

March 16, 2015, 17:52 
Regardless

#9 
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Tali Neuman
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Regardless the set of variables I use (conservative, primitive, ets), for each of them, eigenvectors will have a one degree of freedom (for the 1D case) where I can select freely their value. Am I right?


March 16, 2015, 18:09 

#10 
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Filippo Maria Denaro
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for each eigenvalue lambda(k) you solve the system
[AI lambda(k)]*r(k) = 0 therefore you see that one scalar component is free 

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