Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The need for accuracy and smoothness in numerical simulations

Carl Christian Kjelgaard Mikkelsen, Lorién López-Villellas

TL;DR

The paper analyzes the use of Richardson extrapolation to estimate discretization error in constrained molecular dynamics and related differential-algebraic systems. It derives the asymptotic error framework, defines Richardson estimates, and demonstrates how to extract the dominant error order from data, using both smooth and non-smooth test problems. Empirically, it shows successful use when a valid asymptotic expansion exists (e.g., a howitzer shell model with Euler integration) but also uncovers failures in MD simulations with non-smooth force fields or insufficient solver accuracy, underscoring the need for sufficient smoothness and precision. The work provides practical guidance and diagnostic criteria for applying error estimation in constrained dynamics and time-integration settings, with code and data available for replication.

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.

The need for accuracy and smoothness in numerical simulations

TL;DR

The paper analyzes the use of Richardson extrapolation to estimate discretization error in constrained molecular dynamics and related differential-algebraic systems. It derives the asymptotic error framework, defines Richardson estimates, and demonstrates how to extract the dominant error order from data, using both smooth and non-smooth test problems. Empirically, it shows successful use when a valid asymptotic expansion exists (e.g., a howitzer shell model with Euler integration) but also uncovers failures in MD simulations with non-smooth force fields or insufficient solver accuracy, underscoring the need for sufficient smoothness and precision. The work provides practical guidance and diagnostic criteria for applying error estimation in constrained dynamics and time-integration settings, with code and data available for replication.

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.
Paper Structure (14 sections, 2 theorems, 34 equations, 10 figures, 1 table)

This paper contains 14 sections, 2 theorems, 34 equations, 10 figures, 1 table.

Table of Contents

  1. Motivation
  2. Theory
  3. Elementary examples
  4. Integration of a function that is everywhere smooth.
  5. Integration of a function that is smooth in all but one point.
  6. Practical examples
  7. A successful application of the theory
  8. Example: Identify the shells fired by a howitzer.
  9. An unsuccessful application of the theory
  10. The difference between success and failure
  11. The need for sufficient accuracy
  12. The need for sufficient smoothness
  13. Conclusion
  14. Acknowledgments

Key Result

theorem 1

If $E_h$ satisfies equation equ:aex, then

Figures (10)

  • Figure 1: The behavior of $F_h$, $E_h$ and $R_h$ for a method with $(p,q) = (2,4)$.
  • Figure 2: The evolution $F_h$ and the accuracy of $R_h$ for a method with $(p,q) = (\frac{3}{2},2)$.
  • Figure 3: The evolution of Richardson's fraction corresponding to the maximum range of 3 different shells fired from the D-20 howitzer.
  • Figure 4: The evolution of the kinetic and potential energy of a system as the number of time-steps used to cover $1$ ps of real time.
  • Figure 5: The evolution of the kinetic and potential energy of a system as the number of time-steps used to cover $1$ ps of real time.
  • ...and 5 more figures

Theorems & Definitions (4)

  • theorem 1
  • proof
  • theorem 2
  • proof