Numerical approximation of statistical solutions of scalar conservation laws

TL;DR

This paper introduces efficient numerical algorithms combining finite volume methods with Monte Carlo techniques to approximate statistical solutions of scalar conservation laws, demonstrating convergence and efficiency improvements.

Contribution

The paper develops and proves convergence of combined finite volume and Monte Carlo algorithms, highlighting the efficiency gains of multi-level Monte Carlo over standard methods.

Findings

Both methods accurately compute multi-point statistical quantities.

Multi-level Monte Carlo significantly outperforms standard Monte Carlo in efficiency.

Algorithms converge to the entropy statistical solution.

Abstract

We propose efficient numerical algorithms for approximating statistical solutions of scalar conservation laws. The proposed algorithms combine finite volume spatio-temporal approximations with Monte Carlo and multi-level Monte Carlo discretizations of the probability space. Both sets of methods are proved to converge to the entropy statistical solution. We also prove that there is a considerable gain in efficiency resulting from the multi-level Monte Carlo method over the standard Monte Carlo method. Numerical experiments illustrating the ability of both methods to accurately compute multi-point statistical quantities of interest are also presented.