Conservative finite volume scheme for first-order viscous relativistic hydrodynamics

We present the first conservative finite volume numerical scheme for the causal, stable relativistic Navier-Stokes equations developed by Bemfica, Disconzi, Noronha, and Kovtun (BDNK). BDNK theory has arisen very recently as a promising means of incorporating entropy-generating effects (viscosity, heat conduction) into relativistic fluid models, appearing as a possible alternative to the so-called M\"uller-Israel-Stewart (MIS) theory successfully used to model quark-gluon plasma. The major difference between the two lies in the structure of the system of PDEs: BDNK theory only has a set of conservation laws, whereas MIS also includes a set of evolution equations for its dissipative degrees of freedom. The simpler structure of the BDNK PDEs in this respect allows for rigorous proofs of stability, causality, and hyperbolicity in full generality which have as yet been impossible for MIS. To capitalize on these advantages, we present the first fully conservative multi-dimensional fluid solver for the BDNK equations suitable for physical applications. The scheme includes a flux-conservative discretization, non-oscillatory reconstruction, and a central-upwind numerical flux, and is designed to smoothly transition to a high-resolution shock-capturing perfect fluid solver in the inviscid limit. We assess the robustness of our new method in a series of flat-spacetime tests for a conformal fluid, and provide a detailed comparison with previous approaches of Pandya & Pretorius (2021).