An asymptotic-preserving reduced-order method for parametrised rarefied gas flow by proper generalised decomposition

Abstract

Modelling rarefied gas flow using the Boltzmann equation is vital in many areas. Due to the high dimensionality and coexistence of multiple characteristic scales, conventional solution strategies to this equation incur prohibitively high computational costs and are inadequate for rapid response in engineering design simulations. Based on proper generalised decomposition (PGD), we propose an \textit{a priori}, asymptotic-preserving reduced-order method to solve the high-dimensional, parametrised Shakhov kinetic model equation. The method reduces the original problem to a few low-dimensional problems by formulating separated representations for the low-rank solution, thereby mitigating the curse of dimensionality. To capture the hydrodynamic asymptotics, we incorporated solutions of some synthetic equations into the PGD algorithm. This treatment allows the PGD solver to automatically reduce to a macroscopic solver for the Navier-Stokes equations, whose solution naturally exhibits low-rank structure. By treating the rarefaction parameter as an additional coordinate, a parametrised solution can be computed once and for all over the entire range of rarefaction, enabling fast multiple queries to any points in the parameter space. Numerical examples are presented to demonstrate the capability of the method to simulate rarefied gas flow with certain accuracy and a significant reduction in computational costs.