2.2 SRHD as a hyperbolic system of conservation laws

An important property of system (5View Equation) is that it is hyperbolic for causal EOS [8Jump To The Next Citation Point]. For hyperbolic systems of conservation laws, the Jacobians ∂Fi (u)∕∂u 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.

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