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. Because of the high dimensionality and coexistence of multiple characteristic scales, conventional solution strategies for 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 a priori, asymptotic-preserving reduced-order method for the high-dimensional parameterised Shakhov kinetic model equation. Separated representations reduce the kinetic problem to a set of low-dimensional problems. We incorporate synthetic equations into the PGD iteration so that, in the hydrodynamic limit, the iteration reduces to solving the Navier-Stokes equations, whose solution naturally exhibits a low-rank structure. Furthermore, treating the rarefaction parameter as an additional coordinate yields a solution over the whole range of rarefaction in one offline computation. It enables fast online queries to any point in the parameter space. Numerical examples using this method show reasonable precision and a significant reduction in computational cost.

arXiv commentarXiv admin note: substantial text overlap with arXiv:2505.19555