Computing statistical solutions of a Mach 2000 astrophysical jet

Abstract

The multi-dimensional compressible Euler equations admit non-unique entropy solutions in turbulent regimes, and extreme-Mach astrophysical flows are a natural setting in which this breakdown of deterministic well-posedness becomes computationally visible. We compute statistical solutions of a Mach~2000 astrophysical jet, defined as the pushforward of an initial probability measure through a vectorial lattice Boltzmann method, by Monte Carlo sampling with $M=1000$ realizations on grids of up to $12.8$ million cells. Under mesh refinement the individual realizations diverge pathwise, while the statistical solution converges: Wasserstein distances of the one- and two-point marginals, the ensemble mean, and the ensemble standard deviation all exhibit stable positive convergence rates. A spatially resolved analysis along the jet axis traces this dichotomy to the structure of the one-point laws, which are numerically Dirac in the undisturbed core, skewed in the sheared turbulent regions, and intermittent two-state mixtures at the random leading front. We conclude that the computed statistical solution is non-Dirac and remains stable in the extreme compressible regime, in which no strong solution is expected to exist.

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