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Implicit discretization schemes for Langevin dynamics

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We explore here several numerical schemes for Langevin dynamics in the
general implicit discretization framework of the Langevin/implicit-Euler
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, MID1, 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, MID1 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. MID1 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 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 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.

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