A new front tracking scheme for the ultra-relativistic Euler equations

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure p, the spatial part u∈R3 of the dimensionless four-velocity and the particle density n. Two schemes for these equations are presented in one space dimension, namely a front tracking and a cone-grid scheme. A new front tracking technique for the ultra-relativistic Euler equations is introduced, which gives weak solutions. The front tracking method is based on piecewise constant approximations to Riemann solutions, called front tracking Riemann solutions, where continuous rarefaction waves are approximated by finite collections of discontinuities, so-called non-entropy shocks. This method can be used for analytical as well as for numerical purposes. A new unconditionally stable cone-grid scheme is also derived in this paper, which is based on the Riemann solution for the ultra-relativistic Euler equations. Both schemes are compared by two numerical examples, where explicit solutions are known.