A Riemann problem, named after Bernhard Riemann, consists of a conservation law together with a piecewise constant data having a single discontinuity. The Riemann problem is very useful for the understanding of hyperbolic partial differential equations like the Euler equations because all properties like Shocks, Rarefaction waves appear as characteristics in the solution. As well it gives an exact solution to complicated, non-linear equations like the Euler equations.
In numerical analysis Riemann problems appear in a natural way in finite volume methods for the solution of equation of conservation laws due to the discreteness of the grid. For that it is widely used in computational fluid dynamics and in MHD simulations. In these fields Riemann problems are calculated using Riemann solvers.