The Piecewise Parabolic Method for Multidimensional Relativistic Fluid   Dynamics
  We present an extension of the Piecewise Parabolic Method to special relativistic fluid dynamics in multidimensions. The scheme is conservative, dimensionally unsplit, and suitable for a general equation of state. Temporal evolution is second-order accurate and employs characteristic projection operators; spatial interpolation is piece-wise parabolic making the scheme third-order accurate in smooth regions of the flow away from discontinuities. The algorithm is written for a general system of orthogonal curvilinear coordinates and can be used for computations in non-cartesian geometries. A non-linear iterative Riemann solver based on the two-shock approximation is used in flux calculation. In this approximation, an initial discontinuity decays into a set of discontinuous waves only implying that, in particular, rarefaction waves are treated as flow discontinuities. We also present a new and simple equation of state which approximates the exact result for the relativistic perfect gas with high accuracy. The strength of the new method is demonstrated in a series of numerical tests and more complex simulations in one, two and three dimensions. 
A. Mignone, T. Plewa and G. Bodo