1

I'm studying hyperbolic conservation laws. I went through LeVeque's book [1], chapters 7 and 8. It gives a very good description of shock and rarefaction waves construction for non-linear systems. However I'm not being able to relate it with Riemann invariants. I'm referring Smoller's book [2] for the same.

  1. I know that in case of a shock wave, along the $k$th eigenvector (integral curve) all the components of $u$ remain constant except the $k$th component (strictly hyperbolic systems).

  2. And now the Riemann invariant $w$ is given by $$\nabla_u w \cdot r_k = 0$$ (Does it mean that the Gradient of $w$ with respect to $u$ along the k-eigenvector of the system is zero?)

Are the two things above related in some way? If I'm completely wrong, I would be very glad if you could recommend some references to start with.

[1] Randall J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi:10.1007/978-3-0348-8629-1

[2] Joel Smoller, Shock Waves and Reaction—Diffusion Equations, Springer, 1994. doi:10.1007/978-1-4612-0873-0

EditPiAf
  • 21,328
Black Heart
  • 161
  • I'd recommend Chap. 2 of the book Numerical Approximation of Hyperbolic Systems of Conservation Laws by E Godlewski and P-A Raviart doi:10.1007/978-1-4612-0713-9 – EditPiAf Jan 17 '20 at 14:18
  • As I recall the book Dafermos - Hyperbolic Conservation Laws in Continuum Physics, 2016, has a few things connected to your question. Some mention of Riemman invariants, rarefaction and shock waves and their connections you could find in the answer of this question https://math.stackexchange.com/questions/3208503/solutions-of-a-pde-problem-given-in-the-riemann-invariant-diagonal-form – Mark May 18 '20 at 18:26

0 Answers0