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.
In numerical analysis, Riemann problems appear in a natural way in finite volume methods for the solution of conservation law equations 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.
As a simple example, we investigate the properties of the one-dimensional Riemann problem in gas dynamics, with initial conditions given by
where x = 0 separates two different states, together with the linearised gas dynamic equations (see gas dynamics for derivation).
where we can assume without loss of generality . We can now rewrite the above equations in a conservative form:
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