High-Order Residual Distribution Schemes for Steady 1D Relativistic Hydrodynamics

An important goal in astrophysics is to model phenomena such as the gravitational collapse of stars and accretion onto black holes. Under the assumption that the spacetime metric remains fixed on the time scales of fluid motion, the relevant physics can be modeled by the equations of relativistic hydrodynamics. These equations form a system of hyperbolic balance laws that are strongly nonlinear and exhibit shock formation. In recent years several types of numerical methods have been developed for relativistic hydrodynamics; perhaps the most successful have been high-resolution shock-capturing schemes based either on Godunov, ENO, or central schemes. We present in this work some preliminary results on an alternative approach based on residual distribution schemes. The main attraction of these methods is that they can be made high-resolution shock-capturing and high-order accurate through the use of compact stencils. In the current work we focus specifically on developing second, fourth, and sixth-order accurate residual distribution schemes for 1D steady flows.