Statistical solutions of hyperbolic systems of conservation laws: numerical approximation

TL;DR

This work develops a numerical framework for statistical solutions of multi-dimensional hyperbolic conservation laws by combining high-resolution finite-volume discretizations with Monte Carlo ensembles. It introduces time-dependent correlation measures to represent joint spatial statistics and proves convergence of ensemble approximations to a dissipative statistical solution under verifiable $L^p$ bounds, weak BV estimates, and a Kolmogorov-type scaling hypothesis. It also establishes a Lax–Wendroff-type consistency and a weak-strong uniqueness result in the dissipative statistical-solution setting, and validates the theory through 2D Euler experiments showing convergence of mean, variance, and multi-point observables such as structure functions. The results provide conditional global existence and offer a practical, stable computational approach for uncertainty quantification in multi-dimensional hyperbolic systems, with room for acceleration via ML-based surrogates.

Abstract

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions, are also presented.

Figures (19)

• Figure 1: The approximate density of the Kelvin--Helmholtz instability \ref{['eq:kh']} with a fixed $\omega\in\Omega$ for different mesh resolutions with the same initial data.The scheme used is an HLL3 flux with a WENO2 reconstruction algorithm. In this experiment, $\epsilon=0.01$ and $T=2$.
• Figure 2: Cauchy rates \ref{['eq:crate']} for the approximate density in the Kelvin--Helmholtz problem \ref{['eq:kh']} for different mesh resolutions. The scheme used is a HLL3 flux with a WENO2 reconstruction algorithm. In this experiment, $\epsilon=0.01$. Here $T=2$.
• Figure 3: The approximate mean (top row) and variance (bottom row) of the density of the Kelvin--Helmholtz instability \ref{['eq:kh']} for mesh resolutions of (from left to right) $128^2$, $256^2$, $512^2$ and $1024^2$ points. The scheme used is an HLL3 flux with a WENO2 reconstruction algorithm. In this experiment, $\epsilon=0.01$. The number of samples used, $M$, was set equal to the resolution $N$, and $T=2$.
• Figure 4: Cauchy rates for the Kelvin--Helmholtz problem.
• Figure 5: Structure functions \ref{['eq:sfnum']} for the Kelvin--Helmholtz instability \ref{['eq:kh']} for different times $T$, exponents $p$ and perturbation sizes $\epsilon$. The scheme used is a HLL3 flux with a WENO2 reconstruction algorithm. At each mesh resolution $N$, $M=N$ samples were used.

Theorems & Definitions (55)

• Definition 2.2
• Example 2.3
• Lemma 2.4
• Theorem 2.5
• Theorem 2.6
• Definition 2.7: Fjordholm, Lanthaler, Mishra FLM17
• Theorem 2.8: Fjordholm, Lanthaler, Mishra FLM17
• Remark 2.9
• Lemma 2.10
• proof