2016/11/13 by Ricardo J. Alonso, Alonso, Ricardo J., Irene M. Gamba +3 · 2 citations
Engineering · Mathematics · #35A22 #45E99 #65C20 #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Radiative Heat Transfer Studies #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.1611.04171
openalex publication_date 2016/11/13 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We develop error estimates for the semi-discrete conservative spectral method\nfor the approximation of the elastic and inelastic space homogeneous Boltzmann\nequation introduced by the authors in citeGT09. In addition we study the\nlong time convergence of such semi-discrete solution to equilibrium Maxwellian\ndistribution that conserves the mass, momentum and energy associated to the\ninitial data. The numerical method is based on the Fourier transform of the\ncollisional operator and a Lagrangian optimization correction that enforces the\ncollision invariants, namely conservation of mass, momentum and energy in the\nelastic case, and just mass and momentum in the inelastic one. We present a\ndetailed semi-discrete analysis on convergence of the proposed numerical method\nwhich includes the L1-L2 theory for the scheme. This analysis allows us\nto present, additionally, convergence in Sobolev spaces and convergence to\nequilibrium for the numerical approximation. The results of this work answer a\nlong standing open problem posed by Cercignani et al. in cite[Chapter 12]CIP\nabout finding error estimates for a numerical scheme associated to the\nBoltzmann equation, as well as showing the semi-discrete numerical solution\nconverges to the equilibrium Maxwellian distribution associated to the initial\nvalue problem.\n