Multilevel Monte Carlo Finite Volume Methods for Random Conservation Laws with Discontinuous Flux

TL;DR

This paper develops and analyzes multilevel Monte Carlo finite volume methods for solving random conservation laws with discontinuous flux, providing convergence rates and demonstrating efficiency gains through numerical experiments.

Contribution

It introduces a novel framework for random conservation laws with discontinuous flux and establishes convergence and efficiency of multilevel Monte Carlo methods for these problems.

Findings

Multilevel Monte Carlo methods outperform single-level in efficiency.

Convergence rates are established for the proposed numerical methods.

Numerical experiments validate the theoretical results.

Abstract

We consider conservation laws with discontinuous flux where the initial datum, the flux function, and the discontinuous spatial dependency coefficient are subject to randomness. We establish a notion of random adapted entropy solutions to these equations and prove well-posedness provided that the spatial dependency coefficient is piecewise constant with finitely many discontinuities. In particular, the setting under consideration allows the flux to change across finitely many points in space whose positions are uncertain. We propose a single- and multilevel Monte Carlo method based on a finite volume approximation for each sample. Our analysis includes convergence rate estimates of the resulting Monte Carlo and multilevel Monte Carlo finite volume methods as well as error versus work rates showing that the multilevel variant outperforms the single-level method in terms of efficiency. We present numerical experiments motivated by two-phase reservoir simulations for reservoirs with varying geological properties.