System of conservation laws ($U_t+{\partial }_{x}F(U)=0$)

A system of conservation laws evolves a vector of conserved variables $U$ through a flux vector $F(U)$. For a smooth solution it has quasilinear form $U_t+A(U)U_x=0$, where $A(U)=DF(U)$ is the flux Jacobian.

A first-order quasilinear system has the form $U_t+A(U,x,t)U_x=S(U,x,t)$: derivatives of the unknown enter linearly, while their coefficient matrix may depend on the unknown itself.

A quasilinear system is hyperbolic where its principal coefficient matrix has real eigenvalues and enough independent eigenvectors. The eigenvalues are its characteristic speeds.

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 $f^{\prime }(u)$.