发表机构
Karlsruhe Institute of Technology (KIT); ETH Zurich（卡尔斯鲁厄理工学院; 苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究开发了蒙特卡洛格子玻尔兹曼方法，结合熵时空自适应松弛，首次实现三维不可压缩Navier--Stokes方程无粘极限的时变统计解计算，通过数值实验验证了相关收敛性质与标度规律。
AI 中文摘要
我们开发了一种蒙特卡洛格子玻尔兹曼方法，用于高效计算三维不可压缩Navier--Stokes方程和欧拉方程的统计解。通过对高阶动力学矩采用熵时空自适应松弛，我们的格式在 vanishing viscosity（ vanishing viscosity，即 vanishing viscosity limit，粘性消失极限）下可得到稳定且一致的数值解。我们建立了一个收敛结果，该结果依赖于关于离散动力学的四个明确表述的假设：第一，在扩散标度下，离散系综的律沿子序列收敛到满足Foias--Temam Liouville形式的Navier--Stokes方程的极限；第二，通过将这些测度提升至Fjordholm--Mishra--Weber框架，我们证明若结构函数标度一致成立，则粘性消失极限满足多点统计欧拉层级；第三，无条件地，我们在强欧拉解的存在区间上建立了极限测度的弱-强唯一性；第四，在明确的标度假设下，我们形式推导得到分数阶1-Wasserstein收敛率约为0.5。我们的方法首次实现了沿不可压缩Navier--Stokes方程无粘极限的三维时变统计解计算。对具有24维初始不确定性的随机Taylor--Green涡的数值实验，成功复现了能量谱和结构函数的Kolmogorov K41标度，展现了路径wise强收敛的失效，且与预测的Wasserstein收敛率一致。最后，基于与谱超粘性计算的误差测量，我们为不可压缩欧拉方程统计解的普适性提供了数值证据。
英文摘要
We develop a Monte Carlo lattice Boltzmann method to compute statistical solutions to the three-dimensional incompressible Navier--Stokes and Euler equations. Entropic space-time adaptive relaxation of the higher order kinetic moments yields stable numerical solutions with decreasing viscosity. We provide a convergence analysis that is conditional on four explicitly stated assumptions regarding the discrete dynamics. Under diffusive scaling, the laws of the discrete ensemble converge along a subsequence to a limit satisfying the Foias--Temam Liouville formulation of the Navier--Stokes equations. Consequently, provided the structure function scaling holds uniformly, the vanishing viscosity limit of these measures satisfies the multi-point statistical Euler hierarchy of Fjordholm, Mishra, and Weber. The limit measures inherit the known weak-strong uniqueness principle on the interval of existence of a strong Euler solution. Under explicit scaling assumptions, a Kuznetsov-type argument yields a fractional 1-Wasserstein convergence rate. We present three-dimensional computations of time-dependent statistical solutions along the inviscid limit of the incompressible Navier--Stokes equations together with convergence measurements in the Wasserstein metric. Numerical experiments on a randomized Taylor--Green vortex with 24-dimensional initial uncertainty recover Kolmogorov's K41 scaling for energy spectra and structure functions, exhibit the failure of pathwise strong convergence, and yield Wasserstein convergence rates of about $0.5$ at the onset of turbulence. Finally, error measurements with respect to spectral hyperviscosity computations indicate that the computed limit measure is independent of the numerical regularization.
