Computing weak-strong uniqueness of a Mach 2000 astrophysical jet

2026-05-24

AI Summary

This work addresses the non-uniqueness of entropy solutions to the multidimensional Euler equations in astrophysical jets at extreme Mach numbers (Ma ≈ 2000), where turbulent dynamics render traditional deterministic approaches inadequate for capturing statistical behavior. The authors propose a statistical solution framework based on the vector lattice Boltzmann method (VLBM), leveraging large-scale Monte Carlo sampling and high-resolution simulations—up to 3.2 million grid points and 1,000 realizations—to demonstrate, for the first time in such highly compressible flows, the stability of statistical solutions. While individual realizations diverge in L¹ due to chaotic shear-layer dynamics, their empirical measures converge in Wasserstein distance to a unique limiting distribution at approximately order 0.5. The approach integrates CUDA-optimized kernels, memory-mapped streaming postprocessing, and Cauchy rate analysis, offering numerical evidence pertinent to the weak–strong uniqueness problem.

Abstract

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.

Problem

• weak-strong uniqueness
• astrophysical jet
• extreme Mach number
• non-unique entropy solutions
• statistical solution

Innovation

• statistical solution
• vectorial lattice Boltzmann method
• weak-strong uniqueness
• extreme Mach flow
• Wasserstein convergence