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?
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.),
Let me know if you have any more questions!
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