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-   -   Linear degenerate and genenuiely nonlinear??? (https://www.cfd-online.com/Forums/main/4140-linear-degenerate-genenuiely-nonlinear.html)

 Paul November 23, 2001 09:03

Linear degenerate and genenuiely nonlinear???

In any elementary books about the conservation laws, we will come across the definitions of linear degenerate and genenuiely nonlinear waves. However, I never find the meaning of these two concepts, except the mathematic expressions about the eigenvalues and eigenvectors. Does any one know the meaning of these two guyes?

 Axel Rohde November 23, 2001 10:47

Re: Linear degenerate and genenuiely nonlinear???

For further study on this topic, I highly recommend Ami Harten's original paper on TVNI (Total Variation Non-Increasing), which was later abbreviated to TVD (Total Variation Diminishing).

"High Resolution Schemes for Hyperbolic Conservation Laws" (Ami Harten)

Journal of Computational Physics, Volume 49, p.357-393, 1983

On p.374 it talks about the two waves:

"We consider here systems of conservation laws where the characteristic fields are either genuinely nonlinear (R a <> 0) or linearly degenerate (R a = 0). The waves of a genuinely non-linear field are either shocks or rarefaction waves, depending whether the waves are convergent or divergent. The waves of a linearly degenerate field are exclusively contact discontinuities."

In the above, 'R' is the matrix of right eigenvectors and 'a' is the vector of eigenvalues.

The above paper is written by a mathematician and in my opinion is pretty hard to follow. If you want to read this stuff in 'plain English', then download my dissertation, where I talk about these concepts in detail (TVD, Euler Equations, Eigensytem, etc.),

www.cfd4pc.com/papers.htm

Let me know if you have any more questions!

Axel

 Mukkarum December 26, 2013 01:27

Dear Axel Rohde

I need your dissertation where you talk about these concept in detail (TVD, Euler Equations, Eigen sytem, etc.)

Thanks

 SangeethCFD March 22, 2016 02:32

Dear Paul,

You may find some explanation in the book titled " I do like CFD,Vol 1", pages 42-43.
Below, I share the link of the same: