Monte Carlo simulations of the randomly forced Burgers equation

TL;DR

This paper uses Monte Carlo simulations within a path integral framework to study the behavior of the one-dimensional randomly forced Burgers equation, focusing on shock formation and dynamics.

Contribution

It introduces a Monte Carlo approach to evaluate observables of the Burgers equation in a lattice discretization, providing new insights into shock regularization and structure formation.

Findings

Monte Carlo methods can evaluate structure functions as ensemble averages.

Regularization constraints for shocks are identified for stable simulations.

Insights into shock formation and localized structures are obtained.

Abstract

The behaviour of the one--dimensional random--forced Burgers equation is investigated in the path integral formalism, using a discrete space--time lattice. We show that by means of Monte Carlo methods one may evaluate observables, such as structure functions, as ensemble averages over different field realizations. The regularization of shock solutions to the zero--viscosity limit (Hopf-eq.) eventually leads to constraints on lattice parameters, required for the stability of the simulations. Insight into the formation of localized structures (shocks) and their dynamics is obtained.