AI 中文总结
针对高维参数化沙霍夫动力学模型方程，基于适当广义分解提出渐近保持降阶方法，通过制定低秩解分离表示简化问题，纳入综合方程解捕捉渐近性，可一次性计算参数化解，降低计算成本并保证一定精度。
AI 中文摘要
使用玻尔兹曼方程对稀薄气体流动进行建模在许多领域至关重要。由于其高维度和多特征尺度共存，传统求解策略计算成本过高且不适用于工程设计模拟中的快速响应。基于适当广义分解（PGD），我们提出一种先验的、渐近保持降阶方法来求解高维参数化沙霍夫动力学模型方程。该方法通过为低秩解制定分离表示将原问题简化为几个低维问题，减轻维度诅咒。为捕捉流体动力学渐近性，将一些综合方程的解纳入PGD算法。这样PGD求解器能自动简化为纳维 - 斯托克斯方程的宏观求解器，其解自然具有低秩结构。通过将稀薄参数视为额外坐标，可一次性计算整个稀薄范围内的参数化解，实现对参数空间中任意点的快速多次查询。数值例子表明该方法能以一定精度模拟稀薄气体流动且大幅降低计算成本。
英文摘要
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.
CommentsarXiv admin note: substantial text overlap with arXiv:2505.19555
