On the Applicability of Nonoscillatory Hyperbolic Solvers to Splitting-Method Solutions of the Boltzmann Equation

Abstract

We use various weighted essentially nonoscillatory (WENO) schemes to calculate the advection (free-transport) step of a splitting-method solver for the Boltzmann equation with hard-sphere collisions. The collision integral is calculated using a fast spectral method recently proposed by Filbet and colleagues (2006). For the advection step we consider three different WENO schemes for the purpose of spatial discretization, and use a total variation diminishing Runge-Kutta method for time-stepping. The WENO schemes differ only in terms of the underlying flux scheme (Godunov, Lax-Friedrichs, or Rusanov). We use the one-dimensional shock profile as a test problem, comparing the WENO methods among themselves, and also with a positive flux conservation (PFC) method (Filbet and Russo, 2004) and with Bird’s well-known direct simulation Monte Carlo method. Our results show that the WENO-Godunov method produces accurate and stable shock solutions in good agreement with the Monte Carlo results, whereas solutions obtained with the other two WENO methods and the PFC method display some unphysical features such as density undershoots near the shock front and excessive upstream temperature.