A time-relaxed Monte Carlo method preserving the Navier-Stokes asymptotics

Abstract

In this paper, a new time-relaxed Monte Carlo (TRMC) method is proposed for the inhomogeneous Boltzmann equation. Compared to the standard TRMC scheme, the proposed method performs the same convection operator, however, divides the collision operator by a micro-macro decomposition. The continuous part of the collision operator is constructed based on the first-order Chapman-Enskog expansion and solved by an explicit second-order scheme, while the numerical solution of the rest nonequilibrium part is still provided by the standard TRMC scheme. In this way, the new TRMC method demonstrates the same accuracy as the standard TRMC scheme in the kinetic limit, however, preserves Navier-Stokes asymptotics and the second-order accuracy in the fluid limit. Several numerical cases of inhomogeneous flows, such as the one-dimensional Poiseuille flow, Sod tube flow, the shock wave and two-dimensional hypersonic flow past a cylinder, are calculated and compared with direct simulation Monte Carlo (DSMC) or Navier-Stokes solutions. It is noted that the new TRMC scheme is more accurate and efficient than the standard TRMC and DSMC methods for simulations of multi-scale gas flows.