The simulation of extreme Mach astrophysical flows is traditionally viewed through the lens of deterministic positivity-preserving schemes. However, due to phenomena such as Kelvin--Helmholtz instabilities and shock anomalies, the multi-dimensional Euler equations admit a plethora of non-unique entropy solutions in turbulent regimes. For the first time, we computationally explore the weak-strong uniqueness of a Mach 2000 jet by defining the statistical solution as the pushforward of a probability measure through a vectorial lattice Boltzmann method (VLBM) operator. Utilizing highly optimized CUDA kernels, we compute an ensemble of 1000 Monte Carlo samples across a sequence of unprecedentedly refined spatial grids of up to 3.2 million cells, and subsequently post-process the empirical measures via memory-mapped CPU streaming. We contrast the strong sample-wise $L^1$ error divergence with the convergence of the probability measure in the 1-point Wasserstein distance via empirical Cauchy rates. Our mathematical results demonstrate that while individual flow realizations physically diverge due to chaotic shear-layer instabilities, the macroscopic statistical solution converges to a well-defined limit measure at a rate of 0.5. Conclusively, we provide the first numerical verification of statistical solution stability in the extreme compressible regime.

翻译：极端马赫数天体物理流动的模拟传统上通过确定性保正格式的视角进行研究。然而，由于开尔文-亥姆霍兹不稳定性与激波异常等现象，多维欧拉方程在湍流区域允许存在大量非唯一熵解。我们首次通过将统计解定义为经由矢量格子玻尔兹曼方法算子对概率测度的前推，计算探索了马赫数为2000的喷流弱-强唯一性。利用高度优化的CUDA内核，我们在空间网格精细度空前的序列上——单元总数达320万个——计算了1000个蒙特卡洛样本的系综，随后通过内存映射CPU流处理经验测度。我们对比了强逐样本$L^1$误差发散与基于经验柯西率的1点Wasserstein距离下概率测度的收敛性。数学结果表明：尽管混沌剪切层不稳定性导致个体流动实现物理发散，宏观统计解以0.5的速率收敛至定义明确的极限测度。最终，我们首次提供了极端可压缩区域中统计解稳定性的数值验证。