The need for accuracy and smoothness in numerical simulations

TL;DR

This paper examines the limitations of Richardson extrapolation in molecular dynamics simulations, highlighting the importance of accuracy and smoothness for reliable error estimation in numerical solutions of differential algebraic equations.

Contribution

It identifies conditions necessary for Richardson extrapolation to be effective and demonstrates their violation in GROMACS simulations, providing insights into improving error estimation methods.

Findings

Richardson extrapolation can be unreliable with certain molecular dynamics outputs.

Two key conditions for effective Richardson extrapolation are often not met in GROMACS.

Numerical experiments illustrate the impact of these conditions on error estimation.

Abstract

We consider the problem of estimating the error when solving a system of differential algebraic equations. Richardson extrapolation is a classical technique that can be used to judge when computational errors are irrelevant and estimate the discretization error. We have simulated molecular dynamics with constraints using the GROMACS library and found that the output is not always amenable to Richardson extrapolation. We derive and illustrate Richardson extrapolation using a variety of numerical experiments. We identify two necessary conditions that are not always satisfied by the GROMACS library.