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Normalization of eigenvectors of the Euler equations

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Old   March 16, 2015, 11:34
Default 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.
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Old   March 16, 2015, 14:39
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Quote:
Originally Posted by bubble45 View Post
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.


you can find the answer in the book of LeVeque, remeber you have one degree of freedom in determining the components of the eigenvector
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Old   March 16, 2015, 16:26
Default degree of freedom in determining the components of the eigenvector
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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
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Old   March 16, 2015, 16:49
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Originally Posted by bubble45 View Post
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

have a look through the book, several parts can be useful for you, in particular tha anlysis of Euler equations
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Old   March 16, 2015, 17:28
Default Choice of the number for the degree of freedom component
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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?
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Old   March 16, 2015, 17:32
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Originally Posted by bubble45 View Post
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?
a number if the components comes from the linear equations, otherwise they are functions
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Old   March 16, 2015, 17:44
Default Choosing values for degree of freedom component
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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
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Old   March 16, 2015, 17:48
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Quote:
Originally Posted by bubble45 View Post
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

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
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Old   March 16, 2015, 17:52
Default Regardless
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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?
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Old   March 16, 2015, 18:09
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for each eigenvalue lambda(k) you solve the system

[A-I lambda(k)]*r(k) = 0


therefore you see that one scalar component is free
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