Riemann problem

A Riemann problem, named after Bernhard Riemann, consists of an initial value problem composed of a conservation equation together with piecewise constant data having a single discontinuity. The Riemann problem is very useful for the understanding of equations like Euler conservation equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations.