Computing Statistical Solutions of a Mach 2000 Astrophysical Jet

Jul 16, 2026

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The multi-dimensional compressible Euler equations admit non-unique entropy solutions in turbulent regimes, and extreme-Mach astrophysical flows make this loss of deterministic well-posedness computationally visible. The study computes statistical solutions for a Mach 2000 astrophysical jet as the pushforward of an initial probability measure through a vectorial lattice Boltzmann method using Monte Carlo sampling. With up to 12.8 million-cell grids and 1000 realizations, individual trajectories diverge pathwise while statistical observables converge in Wasserstein metrics.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages29

Reading Time1 h 13 min

Introduction

Motivation and physical/mathematical background

From deterministic benchmarks to statistical solutions

Proposed probabilistic VLBM approach

Statistical vs strong-error convergence

arXiv :2605 .25282v3 [math .NA] 13 Jul 2026 COMPUTING STATISTICAL SOLUTIONS OF A MACH 2000 ASTROPHYSICAL JET∗ STEPHAN SIMONIS† AND GAUTHIER WISSOCQ‡ 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. Key words. statistical solutions, Wasserstein metric, vectorial LBM, high-Mach flows, GPU computing MSC codes. 35L65, 35Q31, 35R60, 65C05, 76M28 1. Introduction. Astrophysical jets at Mach numbers exceeding 103 represent a significant challenge in computational fluid dynamics. The Mach 2000 parameter regime serves two purposes. First, it is directly physically representative of protostellar jets (Herbig–Haro objects) propagating into ultra-cold (T ∼ 10K) molecular clouds, where ambient sound speeds drop below 0 .3km/s, yielding classical Mach numbers in excess of 103 [15] . Second, it serves as a canonical mathematical benchmark for the high-energy limits of the classical Euler equations. By pushing the deterministic breakdown to the extreme compressible limit, we provide a mathematical proxy for the shock-dominated, hyper-turbulent topologies found in more complex relativistic flows, such as active galactic nuclei and gamma-ray bursts [20, 14] . While traditional numerical research utilizes the Mach 2000 jet primarily as a short-duration stress test to validate positivity-preserving limiters [21, 22], the underlying mathematical question of uniqueness in the long-time, fully turbulent regime remains challenging. State-of-the-art high-order frameworks, such as subcell-limited discontinuous Galerkin methods, typically restrict this benchmark to early transient phases (e.g. , t ≤ 0.0015) [24, 23] . Kinetic-theory-derived maximum-principle bounds have recently been used to stabilize deterministic simulations of astrophysical jets up to Mach 800 [25] . In this chaotic regime, foundational convex integration results [1] demonstrate that the multi-dimensional Euler equations may admit infinitely many entropy solutions for the exact same initial data. To recover a well-posed theoretical framework, DiPerna introduced measure-valued solutions [2], which were subsequently ∗ Funding: The work of the first author was supported by the PRIME programme of the German Academic Exchange Service (DAAD), with funding from the Federal Ministry of Research, Technology and Space. The initiation of this work at UZH was supported by a KHYS ConYS grant at KIT in 2024 . †Seminar for Applied Mathematics, ETH Zurich, 8092 Zurich, Switzerland, and Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany ([ssimonis@ethz.ch](ssimonis@ethz.ch), [stephan.simonis@kit.edu](stephan.simonis@kit.edu)). ‡CEA CESTA, BP 2, 33114 Le Barp, France. 2 S. SIMONIS AND G. WISSOCQ refined into statistical solutions [3] . These shift th

FAQs

What problem does the paper target in the Mach 2000 jet regime?

It addresses non-uniqueness of entropy solutions for the compressible Euler equations in fully turbulent, extreme-Mach settings, where deterministic well-posedness breaks down and becomes observable computationally.

How are statistical solutions computed for the Mach 2000 astrophysical jet?

Statistical solutions are obtained by pushing an initial probability measure through a vectorial lattice Boltzmann method, approximated via Monte Carlo sampling with M = 1000 realizations.

What changes under mesh refinement: trajectories or statistical observables?

Individual realizations diverge pathwise under refinement, but statistical observables converge, showing stable positive convergence rates in Wasserstein distances for one- and two-point marginals, plus convergence of ensemble mean and standard deviation.

Jul 16, 2026

Upload by Patrick