Spectral Gap Computations for Linearized Boltzmann Operators

TL;DR

This paper introduces a numerical method to compute spectral gaps of the linearized Boltzmann operator, providing the first quantitative evidence of their existence and approximate values, with convergence analysis and parallel computing implementation.

Contribution

It develops a novel numerical approach using Discontinuous Galerkin projection and constrained minimization to estimate spectral gaps, including convergence proof for integrable cross-sections.

Findings

Numerical evidence of spectral gaps for various cross-sections.

Convergence of the Rayleigh quotient to the true spectral gap.

Implementation of parallel algorithms for large-scale computations.

Abstract

The quantitative information on the spectral gaps for the linearized Boltzmann operator is of primary importance on justifying the Boltzmann model and study of relaxation to equilibrium. This work, for the first time, provides numerical evidences on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a "collision matrix". The original spectral gap problem is then approximated by a constrained minimization problem, with objective function being the Rayleigh quotient of the "collision matrix" and with constraints being the conservation laws. A conservation correction then applies. We also showed the convergence of the approximate Rayleigh quotient to the real spectral gap for the case of integrable angular cross-sections. Some distributed eigen-solvers and hybrid OpenMP and MPI parallel computing are implemented. Numerical results on integrable as well as non-integrable angular cross-sections are provided.