en>MrLinkinPark333: Disambiguated: Bethesda → Bethesda, Maryland, resampling → resampling (statistics)

2014-02-03T04:38:44Z

Disambiguated: Bethesda → Bethesda, Maryland, resampling → resampling (statistics)

New page

A '''Riemann problem''', named after [[Bernhard Riemann]], consists of a [[conservation law]] together with [[piecewise]] constant data having a single [[Discontinuity (mathematics)|discontinuity]]. The Riemann problem
is very useful for the understanding of [[hyperbolic partial differential equation]]s like the [[Euler equations (fluid dynamics)|Euler equations]] because all properties, such as shocks and rarefaction waves, appear as [[Method of characteristics|characteristic]]s in the solution. It also gives an exact solution to some complex nonlinear equations, such as the [[Euler equations (fluid dynamics)|Euler equations]].

In [[numerical analysis]], '''Riemann problems''' appear in a natural way in [[finite volume method]]s 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 [[Computational Magnetohydrodynamics|MHD]] simulations. In these fields Riemann problems are calculated using [[Riemann solver]]s.

==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

: <math>
\begin{bmatrix} \rho \\ u \end{bmatrix} = \begin{bmatrix} \rho_L \\ u_L\end{bmatrix} \text{ for } x \leq 0
\qquad \text{and} \qquad \begin{bmatrix} \rho \\ u \end{bmatrix} = \begin{bmatrix} \rho_R \\ -u_R \end{bmatrix} \text{ for } x > 0
</math>

where ''x'' = 0 separates two different states, together with the linearised gas dynamic equation (see [[gas dynamics]] for derivation)

: <math>
\begin{align}
\frac{\partial\rho}{\partial t} + \rho_0 \frac{\partial u}{\partial x} & = 0 \\[8pt]
\frac{\partial u}{\partial t} + \frac{a^2}{\rho_0} \frac{\partial \rho}{\partial x} & = 0
\end{align}
</math>

we can rewrite the above equation in conservative form <math>U_t + A(U)_x = 0</math>:

: <math>
U = \begin{bmatrix} \rho \\ u \end{bmatrix}, \quad A = \begin{bmatrix} 0 & \rho_0 \\ \frac{a^2}{\rho_0} & 0 \end{bmatrix}
</math>

The [[eigenvalues]] of the system are the [[Method of characteristics|characteristics]] of the system
<math> \lambda_1 = -a, \lambda_2 = a </math>. They give the propagation speed of the medium, including that of any discontinuity, which is the speed of sound here. The corresponding [[eigenvector]]s are

: <math>
\mathbf{e}^{(1)} = \begin{bmatrix} \rho_0 \\ -a \end{bmatrix}, \quad
\mathbf{e}^{(2)} = \begin{bmatrix} \rho_0 \\ a \end{bmatrix}.
</math>

By decomposing the left state <math>u_L</math> in terms of the eigenvectors, we get

: <math>
U_L = \begin{bmatrix} \rho_L \\ u_L \end{bmatrix} = \alpha_1 \begin{bmatrix} \rho_0 \\ -a\end{bmatrix} + \alpha_2 \begin{bmatrix} \rho_0 \\ a \end{bmatrix}.
</math>

Now we can solve for <math>\alpha_1</math> and <math>\alpha_2</math>:

: <math>
\begin{align}
\alpha_1 & = \frac{a \rho_L - \rho_0 u_L}{2a\rho_0} \\[8pt]
\alpha_2 & = \frac{a \rho_L + \rho_0 u_L}{2a\rho_0}
\end{align}
</math>

By doing the same for the right state we get <math>\beta_1</math> and <math>\beta_2</math>. Which is

: <math>
\begin{align}
\beta_1 & = \frac{a \rho_R - \rho_0 u_R}{2a\rho_0} \\[8pt]
\beta_2 & = \frac{a \rho_R + \rho_0 u_R}{2a\rho_0}
\end{align}
</math>

With this, we get the final solution in the domain in between the characteristics, which is

: <math>
U^* = \begin{bmatrix} \rho^* \\ u^* \end{bmatrix} = \beta_1 \begin{bmatrix} \rho_0 \\ -a\end{bmatrix} + \alpha_2 \begin{bmatrix} \rho_0 \\ a \end{bmatrix}
</math>

As this is just a simple example, it still shows the basic properties. Most important the characteristics which decompose the solution into three domains. The propagation speed
of these two equations is equivalent to the propagations speed of the sound.

The fastest characteristic defines the [[Courant–Friedrichs–Lewy condition|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==
{{reflist | colwidth=30em}}
*{{cite book | first=Eleuterio F.| last=Toro| year=1999 | title=Riemann Solvers and Numerical Methods for Fluid Dynamics| publisher=Springer Verlag|location=Berlin | id=ISBN 3-540-65966-8}}
*{{cite book | first=Randall J.| last=LeVeque| year=2004 | title=Finite-Volume Methods for Hyperbolic Problems| publisher=Cambridge University Press|location=Cambridge | id=ISBN 0-521-81087-6}}

==See also==
* [[Computational fluid dynamics]]
* [[Computational magnetohydrodynamics]]
* [[Riemann solver]]

{{DEFAULTSORT:Riemann Problem}}
[[Category:Hyperbolic partial differential equations]]
[[Category:Fluid dynamics]]
[[Category:Computational fluid dynamics]]

Multiple comparisons problem - Revision history

en>MrLinkinPark333: Disambiguated: Bethesda → Bethesda, Maryland, resampling → resampling (statistics)