Exact solution of the 1D Riemann problem in Newtonian and relativistic hydrodynamics

Some of the most interesting scenarios that can be studied in astrophysics, contain fluids and plasma moving under the influence of strong gravitational fields. To study these problems it is required to implement numerical algorithms robust enough to deal with the equations describing such scenarios, which usually involve hydrodynamical shocks. It is traditional that the first problem a student willing to develop research in this area is to numerically solve the one dimensional Riemann problem, both Newtonian and relativistic. Even a more basic requirement is the construction of the exact solution to this problem in order to verify that the numerical implementations are correct. We describe in this paper the construction of the exact solution and a detailed procedure of its implementation.