Riemann problem

A Riemann problem, named after Bernhard Riemann, consists of a conservation law together with a piecewise constant data having a single discontinuity. The Riemann problemis very useful for the understanding of hyperbolic partial differential equation 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.

The Riemann problem in linearized gas dynamics

As a simple example we investigate the properties of the one dimensional Riemann problem in gas dynamics, which is defined by

With this, we get the final solution in the domain in between the characteristics, called "domain of dependence" or "star region", which is
$U^* = egin{bmatrix}
ho^* \ u^* end{bmatrix} = eta_1 egin{bmatrix}
ho_0 \ -aend{bmatrix} + alpha_2 egin{bmatrix}
ho_0 \ a end{bmatrix}$
As this just a simple example, it still shows the basic properties. Most important the characteristics which decompose the solution into three domains. The propagation speedof these two equations is equivalent to the propagations speed of the sound.

The fastest characteristic defines the CFL condition, which sets the restriction for the maximum time step in a computer simulation. Generally as more conservation equations are used, the more characteristics are involved.

References

ee also

* Computational fluid dynamics
* Computational Magnetohydrodynamics
* Riemann solver

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