

2.2 SRHD as a hyperbolic system
of conservation laws
An important property of system (5) is that it is
hyperbolic for causal EOS [8
]. For hyperbolic
systems of conservation laws, the Jacobians
have real eigenvalues and a complete set of
eigenvectors (see Section 9.3). 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 so-called Godunov-type 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 Section
2.3.

