) 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 non-linear
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 subsection.
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Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller http://www.livingreviews.org/lrr-1999-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |