发表机构
Eindhoven University of Technology; Simkinetic B.V.; University of Texas at Austin; Saarland University（埃因霍温理工大学; Simkinetic公司; 德克萨斯大学奥斯汀分校; 萨尔兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出基于φ-散度的Galerkin矩封闭框架，构造兼容近似碰撞算子保证物理性质，结合熵稳定DG方法离散，通过多类数值模拟验证其用于稀薄动力学模拟的鲁棒性与准确性。
AI 中文摘要
本研究引入了一种鲁棒确定性框架，用于近似含二元碰撞的玻尔兹曼方程的解，该框架通过Galerkin方法离散化方程对时间、位置和速度的依赖关系。通过在速度空间中采用基于φ-散度的参数化Galerkin封闭，我们推导了控制流体动力学变量的严格矩方程层级。针对这类封闭本身无法保证真实二元碰撞算子的φ-散度熵耗散这一局限性，我们通过为每个封闭量身定制兼容的近似碰撞算子来恢复该性质。所构造的算子固有地保留了高保真流场模拟所需的基本物理性质，包括伽利略不变性、质量、动量和能量的精确守恒，以及φ-散度熵的严格耗散。此外，我们证明所得的封闭矩系统是对称耗散的，由此产生的柯西问题在时间上局部适定。为将该数学基础转化为高效计算工具，我们采用熵稳定的间断Galerkin（DG）有限元方法离散化位置和时间变量。这种全隐式、熵稳定的时空方法允许时间步长远超典型CFL限制的步长，并可直接计算稳态。该方法的鲁棒性和准确性通过氩气超音速喷管流、通道质量流以及平行壁间热传递的数值模拟得到验证，结果显示其与分析基准、实验测量和随机粒子模拟一致。
英文摘要
This work introduces a robust deterministic framework for approximating solutions of the Boltzmann equation with binary collisions by discretizing their dependence on time, position, and velocity using Galerkin methods. By employing a family of parametric Galerkin closures based on $φ$-divergences in velocity space, we derive rigorous hierarchies of moment equations that govern fluid dynamic variables. Addressing the limitation that these closures alone do not guarantee dissipation of a $φ$-divergence entropy for the true binary collision operator, we restore this property by formulating a compatible approximate collision operator tailored to each closure. This constructed operator intrinsically retains fundamental physical properties essential for high-fidelity flow simulations, including Galilean invariance, exact conservation of mass, momentum, and energy, and strict dissipation of a $φ$-divergence entropy. Furthermore, we show that the resulting closed moment systems are symmetric-dissipative, yielding Cauchy problems that are well-posed locally in time. To translate this mathematical foundation into an efficient computational tool, we discretize the position and time variables with an entropy-stable discontinuous Galerkin (DG) finite element method. The fully implicit, entropy-stable space-time approach enables time steps far beyond typical CFL-limited step sizes and the direct computation of steady states. The robustness and accuracy of the methodology are verified and validated through numerical simulations on the supersonic nozzle flow of argon, mass flow through a channel, and heat transfer between parallel walls, demonstrating agreement with analytical benchmarks, experimental measurements, and stochastic particle simulations.
Comments37 pages, 8 figures, 2 tables, v2: expanded discussion of Fourier transform and conservative spectral methods in the introduction; additional references; acknowledgements added
