Spectral Gap Computations for Linearized Boltzmann Operators
For researchers studying the Boltzmann equation and relaxation to equilibrium, this paper offers the first numerical validation of spectral gaps, though it is an incremental step as it focuses on numerical approximation rather than theoretical proof.
This work provides the first numerical evidence for the existence of spectral gaps in linearized Boltzmann operators and computes approximate values, demonstrating convergence for integrable angular cross-sections.
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.