Fully dissipative relativistic lattice Boltzmann method in two dimensions

TL;DR

This paper introduces a fully dissipative relativistic lattice Boltzmann method in two dimensions for ultra-relativistic fluids, utilizing advanced distribution functions and Gaussian quadratures, validated through multiple tests and compared with theoretical models.

Contribution

It develops a novel 2D relativistic lattice Boltzmann model using third equilibrium distributions and new quadratures, expanding the method's accuracy and applicability.

Findings

Validated with Riemann problem and vortex tests

Transport coefficients match Grad's theory predictions

Higher order expansions improve heat flux and temperature accuracy

Abstract

In this paper, we develop and characterize the fully dissipative Lattice Boltzmann method for ultra-relativistic fluids in two dimensions using three equilibrium distribution functions: Maxwell-J\"uttner, Fermi-Dirac and Bose-Einstein. Our results stem from the expansion of these distribution functions up to fifth order in relativistic polynomials. We also obtain new Gaussian quadratures for square lattices that preserve the spatial resolution. Our models are validated with the Riemann problem and the limitations of lower order expansions to calculate higher order moments are shown. The kinematic viscosity and the thermal conductivity are numerically obtained using the Taylor-Green vortex and the Fourier flow respectively and these transport coefficients are compared with the theoretical prediction from Grad's theory. In order to compare different expansion orders, we analyze the temperature and heat flux fields on the time evolution of a hot spot.