### Abstract

We explore here several numerical schemes for Langevin dynamics in the general implicit discretization framework of the Langevin/implicit-EuIer scheme, LI. Specifically, six schemes are constructed through different discretization combinations of acceleration, velocity, and position. Among them, the explicit BBK method (LE in our notation) and LI are recovered, and the other four (all implicit) are named LIM1, LIM2, MIDI, and MID2. The last two correspond, respectively, to the well-known implicit-midpoint scheme and the trapezoidal rule. LI and LIM1 are first-order accurate and have intrinsic numerical damping. LIM2, MIDI, and MID2 appear to have large-timestep stability as LI but overcome numerical damping. However, numerical results reveal limitations on other grounds. From simulations on a model butane, we find that the nondamping methods give similar results when the timestep is small; however, as the timestep increases, LIM2 exhibits a pronounced rise in the potential energy and produces wider distributions for the bond lengths. MIDI and MID2 appear to be the best among those implicit schemes for Langevin dynamics in terms of reasonably reproducing distributions for bond lengths, bond angles and dihedral angles (in comparison to 1 fs timestep explicit simulations), as well as conserving the total energy reasonably. However, the minimization subproblem (due to the implicit formulation) becomes difficult when the timestep increases further. In terms of computational time, all the implicit schemes are very demanding. Nonetheless, we observe that for moderate timesteps, even when the error is large for the fast motions, it is relatively small for the slow motions. This suggests that it is possible by large timestep algorithms to capture the slow motions without resolving accurately the fast motions.

Original language | English (US) |
---|---|

Pages (from-to) | 1077-1098 |

Number of pages | 22 |

Journal | Molecular Physics |

Volume | 84 |

Issue number | 6 |

DOIs | |

State | Published - Apr 10 1995 |

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### ASJC Scopus subject areas

- Biophysics
- Molecular Biology
- Physical and Theoretical Chemistry
- Condensed Matter Physics

### Cite this

*Molecular Physics*,

*84*(6), 1077-1098. https://doi.org/10.1080/00268979500100761

**Implicit discretization schemes for langevin dynamics.** / Zhang, Guihua; Schlick, Tamar.

Research output: Contribution to journal › Article

*Molecular Physics*, vol. 84, no. 6, pp. 1077-1098. https://doi.org/10.1080/00268979500100761

}

TY - JOUR

T1 - Implicit discretization schemes for langevin dynamics

AU - Zhang, Guihua

AU - Schlick, Tamar

PY - 1995/4/10

Y1 - 1995/4/10

N2 - We explore here several numerical schemes for Langevin dynamics in the general implicit discretization framework of the Langevin/implicit-EuIer scheme, LI. Specifically, six schemes are constructed through different discretization combinations of acceleration, velocity, and position. Among them, the explicit BBK method (LE in our notation) and LI are recovered, and the other four (all implicit) are named LIM1, LIM2, MIDI, and MID2. The last two correspond, respectively, to the well-known implicit-midpoint scheme and the trapezoidal rule. LI and LIM1 are first-order accurate and have intrinsic numerical damping. LIM2, MIDI, and MID2 appear to have large-timestep stability as LI but overcome numerical damping. However, numerical results reveal limitations on other grounds. From simulations on a model butane, we find that the nondamping methods give similar results when the timestep is small; however, as the timestep increases, LIM2 exhibits a pronounced rise in the potential energy and produces wider distributions for the bond lengths. MIDI and MID2 appear to be the best among those implicit schemes for Langevin dynamics in terms of reasonably reproducing distributions for bond lengths, bond angles and dihedral angles (in comparison to 1 fs timestep explicit simulations), as well as conserving the total energy reasonably. However, the minimization subproblem (due to the implicit formulation) becomes difficult when the timestep increases further. In terms of computational time, all the implicit schemes are very demanding. Nonetheless, we observe that for moderate timesteps, even when the error is large for the fast motions, it is relatively small for the slow motions. This suggests that it is possible by large timestep algorithms to capture the slow motions without resolving accurately the fast motions.

AB - We explore here several numerical schemes for Langevin dynamics in the general implicit discretization framework of the Langevin/implicit-EuIer scheme, LI. Specifically, six schemes are constructed through different discretization combinations of acceleration, velocity, and position. Among them, the explicit BBK method (LE in our notation) and LI are recovered, and the other four (all implicit) are named LIM1, LIM2, MIDI, and MID2. The last two correspond, respectively, to the well-known implicit-midpoint scheme and the trapezoidal rule. LI and LIM1 are first-order accurate and have intrinsic numerical damping. LIM2, MIDI, and MID2 appear to have large-timestep stability as LI but overcome numerical damping. However, numerical results reveal limitations on other grounds. From simulations on a model butane, we find that the nondamping methods give similar results when the timestep is small; however, as the timestep increases, LIM2 exhibits a pronounced rise in the potential energy and produces wider distributions for the bond lengths. MIDI and MID2 appear to be the best among those implicit schemes for Langevin dynamics in terms of reasonably reproducing distributions for bond lengths, bond angles and dihedral angles (in comparison to 1 fs timestep explicit simulations), as well as conserving the total energy reasonably. However, the minimization subproblem (due to the implicit formulation) becomes difficult when the timestep increases further. In terms of computational time, all the implicit schemes are very demanding. Nonetheless, we observe that for moderate timesteps, even when the error is large for the fast motions, it is relatively small for the slow motions. This suggests that it is possible by large timestep algorithms to capture the slow motions without resolving accurately the fast motions.

UR - http://www.scopus.com/inward/record.url?scp=0012579993&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0012579993&partnerID=8YFLogxK

U2 - 10.1080/00268979500100761

DO - 10.1080/00268979500100761

M3 - Article

AN - SCOPUS:0012579993

VL - 84

SP - 1077

EP - 1098

JO - Molecular Physics

JF - Molecular Physics

SN - 0026-8976

IS - 6

ER -