Application of Central Upwind Scheme for Solving Special Relativistic Hydrodynamic Equations

Abstract

The accurate modeling of various features in high energy astrophysical scenarios requires the solution of the Einstein equations together with those of special relativistic hydrodynamics (SRHD). Such models are more complicated than the non-relativistic ones due to the nonlinear relations between the conserved and state variables. A high-resolution shock-capturing central upwind scheme is implemented to solve the given set of equations. The proposed technique uses the precise information of local propagation speeds to avoid the excessive numerical diffusion. The second order accuracy of the scheme is obtained with the use of MUSCL-type initial reconstruction and Runge-Kutta time stepping method. After a discussion of the equations solved and of the techniques employed, a series of one and two-dimensional test problems are carried out. To validate the method and assess its accuracy, the staggered central and the kinetic flux-vector splitting schemes are also applied to the same model. The scheme is robust and efficient. Its results are comparable to those obtained from the sophisticated algorithms, even in the case of highly relativistic two-dimensional test problems.

A direct Primitive Variable Recovery Scheme for hyperbolic conservative equations: The case of relativistic hydrodynamics

“…One way of solving this system is by using a Newton-Raphson method (cf. [ 6 ]) but this or any other numerical solution to obtain u ( q ( x , t )) will carry an extra error besides the proper numerical error of the FDM or FVM. This procedure also adds a bit of computational processing time since an iteration loop to find the solution needs to be carried out at each cell every time step.…”

“…Also, we obtain first order convergence for all test in at least one resolution. Additionally, we made an experiment following [ 6 ] of a static Gaussian curve in order to estimate the order of convergence of a smooth static profile which, for this case, reaches a convergence value of about 2 in all the tested resolutions for a fixed time step of 0.01. As expected, this means that the important error of the relativistic Sod shock tube test relays on the discontinuities.…”

“…Somehow, we have to find a good approximation for this integral. Moreover, the flux f inside the integral is evaluated on the boundaries x i±1/2 of the grid cell which, numerically speaking, has no sense because we can only approximate the values of the average charges on the midpoint of the finite volume 6 .…”