Flux-Conservative BDNK Hydrodynamics and Shock Regularization

TL;DR

This work develops a first-order flux-conservative formulation of causal BDNK hydrodynamics for zero chemical potential, focusing on 1+1D conformal fluids. It implements a Kurganov–Tadmor flux scheme with constraint-damping and analyzes frame robustness by varying hydrodynamic frames within the causal regime, while monitoring Knudsen proxies. The study demonstrates stable, frame-robust evolution in the frame-robust regime, and shows that viscosity can regularize potential shocks for certain smooth initial data, with convergence validated by systematic tests. The framework provides a practical, extendable tool for simulating relativistic viscous fluids (e.g., quark-gluon plasma, neutron-star matter) and motivates future nonconformal and higher-dimensional investigations.

Abstract

We present a new, first-order, flux-conservative formulation of relativistic viscous hydrodynamics in the BDNK framework, applicable to conformal and nonconformal fluids at zero chemical potential. Focusing on the conformal case in 1+1 dimensions, we numerically solve the equations of motion for two classes of consistent initial data and assess the robustness of the resulting solutions with respect to changing the hydrodynamic frame. Our flux-conservative formulation does not exhibit spurious oscillatory structures within the regime of validity of BDNK theory (the hydrodynamic-frame-robust regime), corresponding to sufficiently small Knudsen numbers. Using this flux-conservative approach, we numerically investigate the potential formation of shocks and their fate in BDNK in 1+1D using smooth initial data known to produce shocks in the relativistic Euler equations. For the type of initial data we consider, we show that the sharp features formed in Euler are prevented by the hyperbolic viscous BDNK equations. The prevention of shock formation we observe occurs in the frame-robust regime of BDNK. This, however, does not preclude the formation of shocks in BDNK for different smooth initial data. The reliability of our results is supported by systematic numerical convergence testing, which is used to assess shock indicators, simulation performance, and the robustness of solutions for different hydrodynamic frames.

Figures (14)

• Figure 1: A snapshot of the hydrodynamic evolution at $t=30$ GeV$^{-1}$ for Gaussian initial conditions and $4\pi \eta/s = 1,3,20$. Here we employ the $F_2$ hydrodynamic frame, with $c_+ = 0.85,$$a_1 = 25/2,$$a_2 = 25/3$. The simulations pictured were performed on a grid of 48,001 cells and a minimum energy baseline of $\varepsilon_0 = 0.1$ GeV$^4$.
• Figure 2: Convergence test using an $L_1$ norm for Gaussian initial conditions and $4\pi \eta/s = 1,3,20$. Here we employ the $F_2$ hydrodynamic frame. Simulations used in the testing had 12,001, 24,001, and 48,0001 cells.
• Figure 3: Three snapshots of the evolution of the Knudsen numbers $\text{Kn}_u$ and $\text{Kn}_T$ at $t=0,30,60$ GeV$^{-1}$, for Gaussian initial conditions and $4\pi \eta/s = 1,3,20$. Here we employ the $F_2$ hydrodynamic frame. The simulations were performed on a grid of 48,001 cells, with a minimum energy density of $\varepsilon_0 = 0.1$ GeV$^4$.
• Figure 4: Three snapshots of the evolution of the Knudsen numbers $\text{Kn}_u$ and $\text{Kn}_T$ at $t=0,30,60$ GeV$^{-1}$, for Gaussian initial conditions and $4\pi \eta/s = 20$. Here we employ the $F_2$ hydrodynamic frame. This simulation used a minimum energy density of $\varepsilon_0 = 3.2$ GeV$^4$.
• Figure 5: Convergence test using an $L_1$ norm for Gaussian initial conditions with a minimum energy density of $\varepsilon_0 = 3.2$ GeV$^4$ and $4\pi \eta/s = 20$. Here we employ the $F_2$ hydrodynamic frame. The simulations used in this testing were performed on a grid of 12,001, 24,0001, and 48,001 cells.