A Lattice Boltzmann Method for relativistic rarefied flows in (2+1) dimensions

https://doi.org/10.1016/j.jocs.2021.101320Get rights and content

Highlights

  • Free streaming simulations beyond abilities of relativistic lattice kinetic schemes.

  • Such simulations are needed for Quark Gluon Plasma and electron ballistic transport.

  • In order to describe such regime we modify existing relativistic kinetic schemes.

Abstract

We propose an extension to recently developed Relativistic Lattice Boltzmann solvers (RLBM), which allows the simulation of flows close to the free streaming limit. Following previous works Ambruş and Blaga (2018), we use product quadrature rules and select weights and nodes by separately discretizing the radial and the angular components.

This procedure facilitates the development of quadrature-based RLBM with increased isotropy levels, thus improving the accuracy of the method for the simulation of flows beyond the hydrodynamic regime.

In order to quantify the improvement of this discretization procedure over existing methods, we perform numerical tests of shock waves in one and two spatial dimensions in various kinetic regimes across the hydrodynamic and the free-streaming limits.

Introduction

Relativistic flows [1], [2], [3], [4], [5], [6], [7] are of great relevance to several research fields, including astrophysics and cosmology [8], [9] and high energy physics, in particular in connection with the study of the quark gluon plasma (QGP) [10]. Relativistic hydrodynamics has also found application in the context of condensed matter physics, particularly for the study of strongly correlated electronic fluids in exotic (mostly 2-d) materials, such as graphene sheets and Weyl semi-metals [11].

The mounting importance of the relativistic hydrodynamic approach for several physics application areas commands the availability of efficient and versatile simulation tools. In the last decade, the Relativistic Lattice Boltzmann method (RLBM) has gained considerable interest in this context. To date, RLBM has been derived and applied in the limit of vanishingly small Knudsen numbers Kn, defined as the ratio between the particles mean free path and a typical macroscopic scale of the flow; available methods are increasingly inaccurate as one increases the value of Kn, moving towards beyond-hydrodynamic regimes. On the other hand, beyond-hydro regimes are very relevant for QGP, especially with regard to their long-time evolution after the hydrodynamic epoch. Furthermore, electron conduction in pure enough materials is almost ballistic, and therefore more attuned to beyond-hydrodynamic descriptions.

The study these systems has been performed in the past as an eremitic expansion of the purely ballistic regime [12], [13]. In this work we propose instead an extension of RLBM that builds on the hydrodynamic regime to further enhance its efficiency in the rarefied gas regime.

The extension of RLBM to the study of rarefied gases has been previously considered in the work by Ambruş and Blaga [14]. Based on off-lattice product-based quadrature rules, their model allow for an accurate description of one-dimensional flows beyond hydrodynamic regimes.

In this work, we extend the RLBM in order to further enhance its efficiency in the rarefied gas regime.

For simplicity, in this paper we consider gases of massless particles in a (2+1) space time, but the same methodologies can be extended to more general equations of state, suitable for fluids consisting of non-zero mass particles in three space dimensions.

This paper is organized as follows: in the first part of Section 2 we review the main concepts of relativistic kinetic theory, which are instrumental for the subsequent description of the Relativistic Lattice Boltzmann Method. In Section 2.3, we dig deeper into the definition of the model, by describing in more detail a momentum space discretization procedure which enables the beyond-hydro capabilities of the scheme. Finally, in Section 3, we present numerical evidence of the capabilities of the scheme, while Section 4 presents our conclusions and prospects of further development.

Section snippets

Model description

In this work we consider a two-dimensional gas of massless particles; we use a (2+1) dimensional Minkowsky space–time, with metric signature ηαβ=diag(+,,). We adopt Einstein’s summation convention over repeated indices; Greek indices denote (2+1) space–time coordinates and Latin indices two-dimensional spatial coordinates. All physical quantities are expressed in natural units, c=kB=1.

Numerical results

Conclusion

In this paper, we have presented a Relativistic Lattice Boltzmann Method for the simulation of gases of ultra-relativistic particles in two spatial dimensions. The method is able to describe free-streaming dynamics (Kn1) as well as hydro-dynamics (Kn1). The simulation of beyond-hydro regimes is enabled by an off-lattice discretization technique of the momentum space, which comes at the price of introducing some amount of numerical diffusivity.

The procedure consists in adopting a product rule

CRediT authorship contribution statement

L. Bazzanini: Validation, Investigation. A. Gabbana: Conceptualization, Methodology, Software, Writing - original draft, Writing - review & editing, Supervision, Project administration. D. Simeoni: Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Visualization. S. Succi: Conceptualization, Methodology, Writing - review & editing, Supervision, Project administration. R. Tripiccione: Conceptualization, Methodology, Writing - review & editing,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank Luciano Rezzolla and Lukas Weih for useful discussions. DS has been supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 765048. SS acknowledges funding from the European Research Council under the European Union’s Horizon 2020 framework programme (No. P/2014-2020)/ERC Grant Agreement No. 739964 (COPMAT). AG would like to thank professor Michael Günther and professor Matthias

Lorenzo Bazzanini was born in 1996 in Ferrara, Italy. He obtained a Bachelor’s degree in Physics at the University of Ferrara in 2018, with a thesis on the Lattice Boltzmann Method. He is about to graduate at the same University with a Master’s degree in Physics, with a thesis on the relativistic extension of the Lattice Boltzmann Method. His main research interests lie in the fields of kinetic theory and computational fluid dynamics.

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  • Cited by (0)

    Lorenzo Bazzanini was born in 1996 in Ferrara, Italy. He obtained a Bachelor’s degree in Physics at the University of Ferrara in 2018, with a thesis on the Lattice Boltzmann Method. He is about to graduate at the same University with a Master’s degree in Physics, with a thesis on the relativistic extension of the Lattice Boltzmann Method. His main research interests lie in the fields of kinetic theory and computational fluid dynamics.

    Alessandro Gabbana is a postdoc at the Eindhoven University of Technology (The Netherlands). He received his PhD-degree for his thesis work “Lattice Boltzmann Methods for Fluid-Dynamics in Relativistic Regimes”, developed within the framework of the HPC-LEAP European Joint Doctorate programme, with awarding institutions University of Ferrara (Italy) and University of Wuppertal (Germany). His research interests include transport phenomena, computational and statistical physics, high performance computing.

    Daniele Simeoni is currently a Ph.D. student within the framework of the STIMULATE European Joint Doctorate program, with joint degree awarding institutions the Bergische Universitaet Wuppertal, Università degli studi di Ferrara and University of Cyprus.

    He got a Bachelor’s degree in Physics at the University of Rome Tor Vergata in 2009, and then graduated in the same University with a Master’s degree in Physics, with a thesis on the theoretical analysis of the deformation of a bacterial droplet in a shearing emulsion.

    His research interests lie in the field of fluid dynamics. He is particularly interested in the use of computational methods (lattice Boltzmann algorithms) to simulate all kinds of microfluidics systems, mainly active matter systems, emulsions, and complex turbulent flows.

    Sauro Succi is currently a Senior Research Executive and Principal Investigator at The Center for Life Nanosciences at La Sapienza of the Italian Institute of Technology. He is a major contributor to the theory and application of the Lattice Boltzmann method.

    Raffaele Tripiccione is a full professor of Physics at the Department of Physics and Earth Sciences of the University of Ferrara. His early scientific work has been in the area of quantum field theory. Over the years, he has become interested in Lattice Gauge Theories and then on the statistical properties of fluids in the turbulent regime. At present he works mainly on the development of Lattice-Boltzmann inspired algorithms for relativistic fluids.

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