Approximations to the Stochastic Burgers Equation

TL;DR

This paper investigates finite difference methods for the stochastic Burgers equation with space-time white noise, revealing that different schemes converge to different limits and providing theoretical explanations and conjectures.

Contribution

It offers the first detailed analysis of how various finite difference schemes behave for the stochastic Burgers equation with white noise, including theoretical insights and conjectures.

Findings

Different schemes converge to different limits as mesh size decreases.

Theoretical explanation for the convergence behavior.

Numerical evidence supporting conjectures for general equations.

Abstract

This article is devoted to the numerical study of various finite difference approximations to the stochastic Burgers equation. Of particular interest in the one-dimensional case is the situation where the driving noise is white both in space and in time. We demonstrate that in this case, different finite difference schemes converge to different limiting processes as the mesh size tends to zero. A theoretical explanation of this phenomenon is given and we formulate a number of conjectures for more general classes of equations, supported by numerical evidence.