Numerical simulations of first-order viscous relativistic hydrodynamics

Abstract

Bemfica, Disconzi, Noronha, and Kovtun (BDNK) formulated the first causal and stable theory of viscous relativistic hydrodynamics to first-order in the gradient expansion, providing rigorous proofs of hyperbolicity and well-posedness of the underlying equations of motion over an explicit range of hydrodynamic frames. Since then, there has been several numerical and analytic studies of the BDNK equations, ranging from astrophysical to holographic applications, which have revealed their promise in modeling relativistic flows when viscous, first-order corrections to ideal hydrodynamics are important. In this thesis, we present numerical solutions of the BDNK equations obtained via finite-difference methods for conformal fluids in $4$D Minkowski spacetime. We consider flows with variations in only one spatial dimension in Cartesian coordinates, and flows constrained to the surface of a geometric sphere of radius $R$. We find both in the Cartesian geometry and on the two-sphere that, for a particular choice of smooth, stationary initial data with a gaussian peak in the energy density and a sufficiently large value of the entropy-normalized shear viscosity, our numerical simulations lose convergence as the solution evolves into a regime where the relative magnitude of the viscous to zeroth-order terms in the stress-energy tensor are $\mathcal{O}(10)$ and growing, while the weak energy condition is strongly violated (with $u_{\mu}u_{\nu}T^{\mu\nu}=\mathcal{O}(-1)$). We present two additional tests of our numerical scheme to Kelvin-Helmholtz-unstable initial data and small, linear fluid perturbations of equilibrium states. We also present a preliminary qualitative comparison between the Euler and BDNK evolution of initial data which, in the inviscid case, eventually evolves into a turbulent regime. Our low-temperature BDNK simulations demonstrate the damping of high-frequency modes in the energy and vorticity densities, preventing the onset of turbulence in the viscous fluid, which, in one of the cases considered, reaches a steady state within the time frame of the simulation.