动力学方程的波粒分解 II：完整 Boltzmann 方程

中文摘要 AI 辅助

将动力学弛豫方程的波粒分解推广到完整的 Boltzmann 方程。一个恒等变换将碰撞项写为向局部 Maxwell 分布的弛豫加上一个守恒余项。沿特征线，在局部动力学视界上的平衡积分定义了波，而其精确补集——粒子——携带余项。波和粒子方程的矩是扩展的 Navier-Stokes 方程，总守恒律和粒子方程构成波粒多尺度方程，这是一个封闭系统，对每个视界都精确，跨越从 Boltzmann 方程到 Navier-Stokes 流体动力学的连续谱，由视界与弛豫之比参数化。一个耦合的隐式波粒迭代求解该系统，在粒子固定的情况下更新守恒变量，并用预测的波和来自宏观预测的端点牵引力更新粒子。在离散相容性假设下，该迭代保持离散 Boltzmann 解为其不动点，其近连续收敛因子受限于粒子输运份额，该份额随该比率指数衰减。对正激波、Couette 流和空腔流，以及高超声速圆柱和 Apollo 流的测试，在稀薄和过渡区与常规迭代方案一致，在连续极限附近与 Navier-Stokes 解一致。在那里，波粒迭代所需的迭代次数比该方案少约三个到四个数量级以上，或者在该方案不收敛的情况下，比其 Shakhov 版本少，并且 Boltzmann 解在同一网格上的壁钟时间成本是 Navier-Stokes 求解器的 1 到 4 倍。

英文摘要

The wave-particle decomposition of kinetic relaxation equations is extended to the full Boltzmann equation. An identity transformation writes the collision term as a relaxation toward the local Maxwellian plus a conservative remainder. Along characteristics, the equilibrium integral over a local kinetic horizon defines the wave, and its exact complement, the particle, carries the remainder. The moments of the wave and particle equations are extended Navier-Stokes equations, and the total conservation law and the particle equation form the wave-particle multiscale equations, a closed system, exact for every horizon, that spans a continuous spectrum from the Boltzmann equation to Navier-Stokes hydrodynamics, parametrized by the horizon-to-relaxation ratio. A coupled implicit wave-particle iteration solves this system, updating the conservative variables with the particle fixed and the particle with the predicted wave and an endpoint traction from the macroscopic prediction. Under discrete compatibility assumptions the iteration preserves the discrete Boltzmann solution as its fixed point, and its near-continuum convergence factor is bounded by a particle transport share decaying exponentially with this ratio. Tests on a normal shock, Couette and cavity flows, and hypersonic cylinder and Apollo flows agree with the conventional iterative scheme in rarefied and transition regimes and with Navier-Stokes solutions near the continuum limit. There the wave-particle iteration needs about three to more than four orders of magnitude fewer iterations than this scheme or, where it did not converge, its Shakhov version, and the Boltzmann solution costs one to four times the wall time of a Navier-Stokes solver on the same mesh.

发表机构

• Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
• Hong Kong University of Science and Technology（香港科技大学）

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