I need an example of hyperbolic but nonstrictly hyperbolic linear conservation law, but I am not getting one example in any books that I am studying.
My main book is LeVeque (1) and the only example that I got on this is not linear and made my mind confused, because not being diagonalizable, how can be hyperbolic? I also has searching in other books and did not get any linear example.
once, following the same book:
Many thanks.
(1) R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi:10.1007/978-3-0348-8629-1
Best Answer
Consider the linear hyperbolic system of conservation laws $U_t + A U_x = 0$, where $A\sim B$ is similar to the matrix $$ B = \begin{pmatrix} 2 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} . $$ The matrix $A$ is therefore diagonalizable with real eigenvalues, but not all eigenvalues are distinct. This system is not strictly hyperbolic.
The system in OP is analyzed in this post.