Conserved schemes with high pressure ratio, high particle density ratio and self-similar method

The paper proposes a numerical investigation for three conserved schemes when applied to the nonlinear ultra-relativistic Euler equations with an initial high pressure ratio and with an initial high particle density ratio. The results show that two of these schemes work efficiently and the third one may present inaccurate results even applied over a very delicate mesh. Several problems of the relativistic Euler system have self-similar solutions which can be solved by more efficient techniques. We also introduce a representation for self-similar ultra-relativistic Euler equations, which can be solved by the proposed schemes in this article and does not need to reconstruct schemes. Numerical tests are given for the one-dimensional shock tube problems to deal with the problem under consideration. These tests display that increasing the order of accuracy does not help much in upgrading the results. It is also indicated that one-dimensional results solved by self-similar ultra-relativistic Euler equations are almost identical to the exact solutions.