Strong Convergence of Integrators for Nonequilibrium Langevin Dynamics`

TL;DR

This paper analyzes the convergence of numerical integrators for Nonequilibrium Langevin Dynamics, highlighting issues with naive schemes and proposing higher-order methods that ensure strong convergence under deforming boundary conditions.

Contribution

The paper introduces and analyzes first and second order strong convergence schemes specifically designed for NELD with deforming boundary conditions, addressing limitations of standard methods.

Findings

Naive schemes lose convergence under deforming boundaries.

Proposed schemes achieve strong convergence of order one and two.

Analysis guides the selection of numerical integrators for NELD.

Abstract

Several numerical schemes are proposed for the solution of Nonequilibrium Langevin Dynamics (NELD), and the rate of convergence is analyzed. Due to the special deforming boundary conditions used, care must be taken when using standard stochastic integration schemes, and we demonstrate a loss of convergence for a naive implementation. We then present several first and second order schemes, in the sense of strong convergence.