An important property of system (5) is that it is hyperbolic for causal EOS [6]. For hyperbolic systems of conservation laws, the Jacobians
have real eigenvalues and a complete set of eigenvectors (see
Section
9.2). Information about the solution propagates at finite velocities
given by the eigenvalues of the Jacobians. Hence, if the solution
is known (in some spatial domain) at some given time, this fact
can be used to advance the solution to some later time (initial
value problem). However, in general, it is not possible to derive
the exact solution for this problem. Instead one has to rely on
numerical methods which provide an approximation to the solution.
Moreover, these numerical methods must be able to handle
discontinuous solutions, which are inherent to nonlinear
hyperbolic systems.
The simplest initial value problem with discontinuous data is
called a Riemann problem, where the one dimensional initial state
consists of two constant states separated by a discontinuity. The
majority of modern numerical methods, the socalled Godunovtype
methods, are based on exact or approximate solutions of Riemann
problems. Because of its theoretical and numerical importance, we
discuss the solution of the special relativistic Riemann problem
in the next subsection.

Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr19993
© MaxPlanckGesellschaft. ISSN 14338351
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