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. 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 computational magnetohydrodynamics simulations. In these fields, Riemann problems are calculated using Riemann solvers.

The Riemann problem in linearized gas dynamics

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 begin{bmatrix} rho u end{bmatrix} = begin{bmatrix} rho_L u_Lend{bmatrix} text{ for } x leq 0qquad text{and} qquad begin{bmatrix} rho u end{bmatrix} = begin{bmatrix} rho_R u_R end{bmatrix} text{ for } x > 0where x = 0 separates two different states, together with the linearised gas dynamic equations (see gas dynamics for derivation). begin{align}frac{partialrho}{partial t} + rho_0 frac{partial u}{partial x} & = 0 [8pt] end{align}where we can assume without loss of generality age 0.We can now rewrite the above equations in a conservative form: U_t + A cdot U_x = 0 where U = begin{bmatrix} rho u end{bmatrix}, quad A = begin{bmatrix} 0 & rho_0 frac{a^2}{rho_0} & 0 end{bmatrix}and the index denotes the partial derivative with respect to the corresponding variable (i.e. x or t).The eigenvalues of the system are the characteristics of the system

mathbf{e}^{(1)} = begin{bmatrix} rho_0 -a end{bmatrix}, quad mathbf{e}^{(2)} = begin{bmatrix} rho_0 a end{bmatrix}.By decomposing the left state u_L in terms of the eigenvectors, we get for some alpha_{1},alpha_{2} U_L = begin{bmatrix} rho_L u_L end{bmatrix} = alpha_1mathbf{e}^{(1)} + alpha_2 mathbf{e}^{(2)} .Now we can solve for alpha_1 and alpha_2: begin{align}alpha_1 & = frac{a rho_L - rho_0 u_L}{2arho_0} [8pt]alpha_2 & = frac{a rho_L + rho_0 u_L}{2arho_0}end{align}Analogously for begin{align}beta_1 & = frac{a rho_R - rho_0 u_R}{2arho_0} [8pt]beta_2 & = frac{a rho_R + rho_0 u_R}{2arho_0}end{align}Using this, in the domain in between the two characteristics t=|x|/a,we get the final constant solution: U_* = begin{bmatrix} rho_* u_* end{bmatrix}

beta_1mathbf{e}^{(1)}+alpha_2mathbf{e}^{(2)}

beta_1 begin{bmatrix} rho_0 -aend{bmatrix} + alpha_2 begin{bmatrix} rho_0 a end{bmatrix}

and the (piecewise constant) solution in the entire domain t>0:

begin{bmatrix} rho(t,x) u(t,x)end{bmatrix}

begin{cases}

U_L, & 0