# Statistical solutions of hyperbolic systems of conservation laws: numerical approximation

## TL;DR

This paper introduces a numerical algorithm combining finite volume methods and Monte Carlo sampling to approximate statistical solutions of hyperbolic conservation laws, with proven convergence and demonstrated properties.

## Contribution

It presents a new computational approach for approximating statistical solutions, integrating high-resolution finite volume methods with Monte Carlo sampling, and proves convergence under certain conditions.

## Key findings

- Algorithm converges to statistical solutions
- Numerical experiments confirm theoretical convergence
- Reveals properties of statistical solutions

## 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.