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Discrete gradients in short-range molecular dynamics simulations

Volker Grimm, Tobias Kliesch, G. R. W. Quispel

TL;DR

The paper develops discrete gradient (DG) integrators tailored for short-range molecular dynamics by formulating MD forces as DGs, ensuring exact energy conservation for conservative dynamics modeled by $y' = J\nabla_y H(y)$. It introduces distance-based DG constructions for pairwise, angle, and dihedral interactions, including new symmetric DGs for dihedral angles, and demonstrates second-order accuracy when DGs are symmetric. Through multiple MD tests (Lennard–Jones dimers, bond angles, dihedrals, butane) and parallel implementations using domain decomposition and MPI, the authors show robust energy preservation, momentum invariants under appropriate conditions, and scalable performance. The method provides a principled, structure-preserving alternative to traditional integrators (e.g., Verlet, implicit midpoint) for true NVE simulations and large-scale MD, with practical guidance for parallel DG evaluation of short-range forces.

Abstract

Discrete gradients (DG) or more exactly discrete gradient methods are time integration schemes that are custom-built to preserve first integrals or Lyapunov functions of a given ordinary differential equation (ODE). In conservative molecular dynamics (MD) simulations, the energy of the system is constant and therefore a first integral of motion. Hence, discrete gradient methods seem to be a natural choice as an integration scheme in conservative molecular dynamics simulations.

Discrete gradients in short-range molecular dynamics simulations

TL;DR

The paper develops discrete gradient (DG) integrators tailored for short-range molecular dynamics by formulating MD forces as DGs, ensuring exact energy conservation for conservative dynamics modeled by . It introduces distance-based DG constructions for pairwise, angle, and dihedral interactions, including new symmetric DGs for dihedral angles, and demonstrates second-order accuracy when DGs are symmetric. Through multiple MD tests (Lennard–Jones dimers, bond angles, dihedrals, butane) and parallel implementations using domain decomposition and MPI, the authors show robust energy preservation, momentum invariants under appropriate conditions, and scalable performance. The method provides a principled, structure-preserving alternative to traditional integrators (e.g., Verlet, implicit midpoint) for true NVE simulations and large-scale MD, with practical guidance for parallel DG evaluation of short-range forces.

Abstract

Discrete gradients (DG) or more exactly discrete gradient methods are time integration schemes that are custom-built to preserve first integrals or Lyapunov functions of a given ordinary differential equation (ODE). In conservative molecular dynamics (MD) simulations, the energy of the system is constant and therefore a first integral of motion. Hence, discrete gradient methods seem to be a natural choice as an integration scheme in conservative molecular dynamics simulations.
Paper Structure (17 sections, 13 theorems, 93 equations, 18 figures, 2 tables)

This paper contains 17 sections, 13 theorems, 93 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Discrete gradients in geometric integration
  3. DG methods in molecular dynamics (MD) simulations
  4. Separable Hamiltonian systems
  5. Discrete gradients for molecular dynamics
  6. Discrete gradients for pairwise forces
  7. Experiment with two Lennard--Jones particles
  8. Discrete gradients for bond angles
  9. Experiment with bond angles
  10. Discrete gradients for dihedral angles
  11. Experiment with butane
  12. Parallelisation of DG methods
  13. Parallelisation of the evaluation of discrete gradients for Lennard--Jones forces with cut-off
  14. Collision of two bodies
  15. Parallelisation of the evaluation of discrete gradients for bonded forces
  16. ...and 2 more sections

Key Result

Proposition 1

$\overline{\nabla}V({\bf u},{\bf u}')$ is a discrete gradient if and only if it is continuous and where $w({\bf u},{\bf u}')$ is a vector-valued function such that where $P_{({\bf u}'-{\bf u})^\perp}$ is the projection on the space perpendicular to $({\bf u}'-{\bf u})$.

Figures (18)

  • Figure 1: Initial conditions for the experiment with two Lennard--Jones particles: the positions and velocities are given on the left-hand side. The first row refers to the particle number, the next three rows to the three coordinates. On the right-hand side, the initial positions are given as a plot made by the open visualization tool (OVITO), cf.Stukowski10.
  • Figure 2: Results of the experiment with two Lennard--Jones particles: the error versus the time step is shown on the left-hand side for the discrete gradient (DG) method, the midpoint rule, and the Verlet scheme. On the right-hand side, the energy is shown over the time span $[0,10]$ for step-size $\tau=0.005$ for all three methods.
  • Figure 3: Sketch of bond angle: From left to right, first the bonds and the bond angle between the atoms $i$,$j$,$k$ is shown. The second sketch shows the distances between the atoms $i$,$j$,$k$ used in the discrete gradient. On the right-hand side, the standard term of the bond angle is followed by the term based on distances, solely.
  • Figure 4: Initial conditions for the experiment with two water-like molecules: the positions and velocities are given on the left-hand side. In the positions section, the first row numbers the particles. The second row assigns molecule numbers. If the number is the same, a bond is added between the particles. The third to fifth row are the coordinates of the positions. In the velocities section, the particle number is followed by velocities. On the right-hand side, the initial configuration is given as a plot by the open visualization tool (OVITO), cf.Stukowski10
  • Figure 5: Results of the experiment with two water-like molecules: the error versus the time step is shown on the left-hand side for the discrete gradient (DG) methods, the midpoint rule, and the Verlet scheme. On the right-hand side, the energy is shown over the time span $[0,10]$ for step-size $\tau=0.005$ for all methods.
  • ...and 13 more figures

Theorems & Definitions (23)

  • Definition 1
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 13 more