A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest.
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.
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.
As a simple example, we investigate the properties of the one-dimensional Riemann problem in gas dynamics (Toro, Eleuterio F. (1999).
Riemann Solvers and Numerical Methods for Fluid Dynamics, Pg 44, Example 2.5) The initial conditions are given by where x = 0 separates two different states, together with the linearised gas dynamic equations (see gas dynamics for derivation).
We can now rewrite the above equations in a conservative form: where and the index denotes the partial derivative with respect to the corresponding variable (i.e. x or t).
The corresponding eigenvectors are By decomposing the left state
: Although this is a simple example, it still shows the basic properties.
Most notably, the characteristics decompose the solution into three domains.
The fastest characteristic defines the Courant–Friedrichs–Lewy (CFL) condition, which sets the restriction for the maximum time step for which an explicit numerical method is stable.
Generally as more conservation equations are used, more characteristics are involved.