[Paper Review] Common Molecular Dynamics Algorithms Revisited

Alexey K. Mazur|arXiv (Cornell University)|Sep 15, 1997

Scientific Research and Discoveries23 references31 citations

TL;DR

This paper re-evaluates the leapfrog integrator in molecular dynamics, demonstrating that its energy conservation and truncation error properties are superior to conventional belief. It identifies interpolation errors in standard energy fluctuation measurements as the root cause of apparent instability, and proposes a corrected method free from such errors, suggesting current step sizes may be unnecessarily small for accurate sampling.

ABSTRACT

The Stormer?Verlet?leapfrog group of integrators commonly used in molecular dynamics simulations has long become a textbook subject and seems to have been studied exhaustively. There are, however, a few striking effects in performance of algorithms which are well known but have not received adequate attention in the literature. A closer view of these unclear observations results in unexpected conclusions. It is shown here that contrary to the conventional point of view, the leapfrog scheme is distinguished in this group both in terms of the order of truncation errors and the conservation of the total energy. In this case the characteristic square growth of fluctuations of the total energy with the step size, commonly measured in numerical tests, results from additional interpolation errors with no relation to the accuracy of the computed trajectory. An alternative procedure is described for checking energy conservation of leapfrog-like algorithms which is free from interpolation errors. Preliminary tests on a representative model system suggest that standard step size values used at present are lower than necessary for accurate sampling.

Motivation & Objective

• To re-express and reassess the performance of the leapfrog integrator in molecular dynamics simulations.
• To identify and resolve the source of misleading energy fluctuation measurements in standard numerical tests.
• To propose a new method for evaluating energy conservation that eliminates interpolation errors.
• To challenge the conventional wisdom on step size selection in molecular dynamics by showing current values may be unnecessarily small.

Proposed method

• Analyzes the leapfrog integrator's truncation errors and energy conservation properties using theoretical and numerical analysis.
• Identifies that apparent energy fluctuations grow quadratically with step size due to interpolation errors, not inherent inaccuracy.
• Proposes an alternative energy evaluation procedure that avoids interpolation by using consistent time-point sampling.
• Employs a representative model system to test the new method and compare energy conservation under standard and corrected procedures.
• Uses numerical experiments to compare energy fluctuations and trajectory accuracy across different step sizes.

Experimental results

Research questions

• RQ1 Why do energy fluctuations in leapfrog integrators appear to grow quadratically with step size in standard numerical tests?
• RQ2 What causes the discrepancy between theoretical energy conservation and observed fluctuations in leapfrog simulations?
• RQ3 How can energy conservation be measured accurately without introducing interpolation errors?
• RQ4 Is the current standard step size in molecular dynamics simulations unnecessarily small for accurate sampling?
• RQ5 What is the true relationship between step size and energy conservation in leapfrog-like algorithms?

Key findings

• The leapfrog integrator exhibits superior truncation error properties and energy conservation compared to other members of the Verlet family.
• The observed quadratic growth of energy fluctuations with step size is an artifact of interpolation errors in standard measurement methods, not a sign of poor accuracy.
• The proposed alternative energy evaluation method eliminates interpolation errors and provides a more accurate assessment of energy conservation.
• Preliminary tests indicate that current standard step sizes in molecular dynamics are lower than necessary for accurate sampling.
• The leapfrog method's performance is better than conventionally assumed, especially when energy conservation is measured correctly.