A numerical exploration of first-order relativistic hydrodynamics

We present the first numerical solutions of the causal, stable relativistic Navier-Stokes equations as formulated by Bemfica, Disconzi, Noronha, and Kovtun (BDNK). For this initial investigation we restrict to plane-symmetric configurations of a conformal fluid in Minkowski spacetime. We consider evolution of three classes of initial data: a smooth (initially) stationary concentration of energy, a standard shock tube setup, and a smooth shockwave setup. We compare these solutions to those obtained with the Muller-Israel-Stewart (MIS) formalism, variants of which are the common tools used to model relativistic, viscous fluids. We find that for the two smooth initial data cases, simple finite difference methods are adequate to obtain stable, convergent solutions to the BDNK equations. For low viscosity, the MIS and BDNK evolutions show good agreement. At high viscosity the solutions begin to differ in regions with large gradients, and there the BDNK solutions can (as expected) exhibit violation of the weak energy condition. This behavior is transient, and the solutions evolve toward a hydrodynamic regime in a way reminiscent of an approach to a universal attractor. For the shockwave problem, we give evidence that if a hydrodynamic frame is chosen so that the maximum characteristic speed of the BDNK system is the speed of light (or larger), arbitrarily strong shockwaves are smoothly resolved. Regarding the shock tube problem, it is unclear whether discontinuous initial data is mathematically well-posed for the BDNK system, even in a weak sense. Nevertheless we attempt numerical solution, and then need to treat the perfect fluid terms using high-resolution shock-capturing methods. When such methods can successfully evolve the solution beyond the initial time, subsequent evolution agrees with corresponding MIS solutions, as well as the perfect fluid solution in the limit of zero viscosity.