Flux Jacobian ($A(U)=DF(U)$)

The flux Jacobian is the Jacobian matrix of the flux in a system of conservation laws. Its eigenvalues are the local characteristic speeds, and its left and right eigenvectors select the corresponding wave variables and directions in state space.

A characteristic speed is an eigenvalue of the flux Jacobian. Along a corresponding characteristic curve, an associated wave variable obeys an ordinary differential equation obtained by projecting the governing system onto the matching left eigenvector.

A kinematic wave is a disturbance whose propagation speed is determined by the local relation between a conserved density and its flux. In a scalar conservation law $u_t+f(u{)}_x=0$, smooth values travel at speed $f^{\prime }(u)$.