3.3.4 The Riemann problem

Modern conservative methods (HRSC schemes) are based on exact or approximate solutions of Riemann problems between contiguous numerical cells

Consider the hyperbolic system of conservation laws in 1D

$\begin{displaymath} \frac{\partial {\bf u}}{\partial t} + \frac{\partial {\bf f}({\bf u})}{\partial x} = 0 \end{displaymath}$

(95)

with initial data ${\bf u}(x,0)={\bf u}_0(x)$

A Riemann problem for system (96) is an initial value problem with discontinuous data, i.e.,

$\begin{displaymath} {\bf u}_0 = \left\{ \begin{array}{ll} {\bf u}_L & \mbox{if $x<0$} \\ {\bf u}_R & \mbox{if $x>0$} \end{array} \right. \end{displaymath}$

(96)

The Riemann problem is invariant under similarity transformations $(x,t) \rightarrow (ax,at)$ , , so that the solution is constant along the straight lines constant and, hence, self-similar

It consists of constant states separated by rarefaction waves (continuous self-similar solutions of the differential system), shock waves and contact discontinuities (Lax 1972).

$\begin{figure}\centerline{\psfig{figure=../FIGURES_DATABASE/riem_short.eps,width=12cm}}\end{figure}$