Lattice Boltzmann Methods for Fluid-Dynamics in Relativistic Regimes

Abstract

In this thesis work we present the algorithmic derivation of a new class of Lattice Boltzmann Methods appropriate for the study of dissipative relativistic fluids. While previous models were restricted to the use of massless particles, implying ultra-relativistic equations of state, this work provides a significant step towards the formulation of a unified relativistic lattice kinetic scheme, covering ideal gases of both massive and near-massless particles, seamlessly bridging the gap between relativistic and low-speed non-relativistic fluid regimes. In a first important application of this novel numerical tool, we present results bringing new insight in the long standing problem of understanding the pathway from relativistic kinetic theory to relativistic hydrodynamics. We conduct an accurate analysis of the relativistic transport coefficients in the relaxation time approximation, providing numerical evidence that the Chapman Enskog expansion correctly relates kinetic transport coefficients and macroscopic hydrodynamics parameters in dissipative relativistic fluid dynamics, confirming recent theoretical results. This analysis, in turn, can be seen as an accurate calibration of this class of numerical solvers, making them suitable to deliver improved physical accuracy in the simulation of realistic systems. To give an example, we present results of simulations solving the Riemann problem for a quark-gluon plasma, showing good agreement with previous results obtained using other solvers present in the literature. As a further application we study the transport properties of electrons in ultra-clean graphene samples, for which a hydrodynamic description is appropriate due to the predominance of electron-electron scattering over electron-phonon interactions. Using appropriate 2D formulations, enriched to describe the effects of the external electrostatic drive, and to capture the interactions with phonons and impurities, we present simulations of laminar flows taking into consideration geometrical setups used in actual experiments. Furthermore, we also consider electronic systems where nonlinear effects start becoming relevant. Basing on extensive numerical simulations, we identify transport parameters which could be used to trigger and observe preturbulent signals in a hydrodynamic region as close as possible to those within reach of current experimental conditions. As a closing note, we remark that the numerical methods described in this thesis work retains the main computational advantages of standard Lattice Boltzmann Methods, offering high amenability to parallelization, that can be exploited to write efficient codes. These aspects are covered in the last chapter of the thesis, in which we summarize the best practices in the development of a performance portable code targeting modern high performance computing accelerators.