Roe linearization and Roe numerical flux

For states $u_L,u_R$ in the open state domain $G$ of Strictly hyperbolic conservation systems and characteristic families, a Roe matrix is a map $\tilde{A}:G\times G\to {\mathbb{R}}^{m\times m}$ such that, for all $u,u_L,u_R\in G$, $\tilde{A}(u,u)=Df(u),\tilde{A}(u_L,u_R)(u_R-u_L)=f(u_R)-f(u_L),$ and $\tilde{A}(u_L,u_R)$ is diagonalizable over $\mathbb{R}$ with real eigenvalues. The map $\tilde{A}$ is also continuous at each diagonal pair $(u,u)$. If $\tilde{A}=R\mathrm{\Lambda }{R}^{-1}$, define $\mathrm{\mid }\tilde{A}\mathrm{\mid }=R\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }{R}^{-1}$ by replacing each diagonal eigenvalue with its absolute value. The Roe flux associated with this matrix is ${\widehat{f}}_{\mathrm{R}\mathrm{o}\mathrm{e}}(u_L,u_R)=\frac{1}{2}(f(u_L)+f(u_R))-\frac{1}{2}\mathrm{\mid }\tilde{A}(u_L,u_R)\mathrm{\mid }(u_R-u_L)\mathrm{.}$ The jump identity is an algebraic conservation property. This definition does not assert entropy admissibility of the approximate wave pattern.