Modelling rarefied gas flow via the Boltzmann equation plays a vital role in many areas. Due to the high dimensionality of this kinetic equation and the coexistence of multiple characteristic scales in the transport processes, conventional solution strategies incur prohibitively high computational costs and are inadequate for rapid response for parametric analysis and optimisation loops in engineering design simulations. This paper proposes an a priori reduced-order method based on the proper generalised decomposition to solve the high-dimensional, parametrised Shakhov kinetic model equation. This method reduces the original problem into a few low-dimensional problem by formulating separated representations for the low-rank solution, as well as data and operators in the equation, thereby overcoming the curse of dimensionality. Furthermore, a general solution can be calculated once and for all in the whole range of the rarefaction parameter, enabling fast and multiple queries to a specific solution at any point in the parameter space. Numerical examples are presented to demonstrate the capability of the method to simulate rarefied gas flow with high accuracy and significant reduction in CPU time and memory requirements.
Su et al. (Mon,) studied this question.