Frequency-domain general synthetic iterative scheme for efficient simulation of oscillatory rarefied gas flows

Abstract

Oscillatory rarefied gas flows are frequently encountered in MEMS, and their efficient numerical simulation remains a major challenge due to the time dependent nature of the problem and the high dimensionality of the Boltzmann kinetic equation. Here, we address this challenge by focusing on the periodic steady state and solving the resulting problem using the frequency domain general synthetic iterative scheme (GSIS). The key idea of GSIS is to simultaneously solve the mesoscopic kinetic equation and the macroscopic synthetic equation. The kinetic equation provides high-order constitutive relations, beyond those given by the Newton law of viscosity and the Fourier law of heat conduction, to the synthetic equation. In turn, the synthetic equation, which converges to the periodic steady state much faster than the kinetic equation, boosts the evolution of the kinetic equation toward the periodic steady state. As a result, super convergence is achieved, together with an asymptotic preserving property that allows the use of coarse spatial grids. The analytical Fourier stability analysis and the Chapman-Enskog expansion, together with challenging numerical simulations, are employed to demonstrate the fast convergence and asymptotic-preserving properties of GSIS, revealing that it can be three orders of magnitude faster than conventional kinetic schemes in near continuum flow regimes.