Constraint evolution in first-order viscous relativistic fluids

TL;DR

This paper analyzes the evolution and stability of constraints in first-order viscous relativistic fluids, demonstrating their well-posedness analytically and numerically, which supports the BDNK reduction's suitability for simulations.

Contribution

It proves the hyperbolic nature of the constraints' evolution equations and validates their numerical stability in flat spacetime simulations.

Findings

Constraints satisfy a strongly-hyperbolic system ensuring proper propagation.

Numerical simulations show stable evolution of constraints with small errors.

BDNK reduction is suitable for accurate numerical evolutions of viscous fluids.

Abstract

Relativistic hydrodynamics provides a solid framework for evolving matter and energy in a wide variety of phenomena. Nevertheless, the inclusion of dissipative effects in realistic scenarios through causal, stable, and well-posed theories still constitutes an open problem. In this paper, we study the evolution of the algebraic and differential constraints stemmed from the first-order reduction proposed by Bemfica, Disconzi, Noronha and Kovtun (BDNK), for proving the local well-posedness of conformally-invariant viscous fluids in Sobolev spaces. First, we show analytically that the whole set of constraints satisfies a homogeneous, strongly-hyperbolic system of equations, ensuring a correct propagation as a consequence of the fluid equations. Motivated by this result, we explore their numerical stability by performing simulations of the BDNK reduction restricted to plane-symmetric configurations, in flat spacetime. We report on different initial data sets initially satisfying the constraints, and whose evolution leads to stable configurations. This result suggests that the proposed reduction by BDNK is suitable for numerical evolutions, keeping the constraints accurate under small numerical errors.