Convergence of a splitting scheme applied to the Ruijgrok–Wu model of the Boltzmann equation

摘要

This paper deals with upwind splitting schemes for the Ruijgrok-Wu model (Physica A 113 (1982) 401–416) of the kinetic theory of rarefied gases in the fluid-dynamic scaling. We prove the stability and the convergence for these schemes. The relaxation limit is also investigated and the limit equation is proved to be a first-order quasi-linear conservation law. The loss of quasi-monotonicity of the present model makes it necessary to give a more careful analysis of its structure. We also obtain global error estimates in the spaces Ws,p for −1⩽s⩽1/p,1⩽p⩽∞ and pointwise error estimates for the approximate solution. The proof naturally uses the framework introduced by Nessyahu and Tadmor (SIAM J. Numer Anal. 29 (1992) 1505–1519) due to the convexity of the flux function.