IMPLICIT DICRETIZATION SCHEMES FOR LANGEVIN DYNAMICS

We explore here several numerical schemes for Langevin dynamics in the general implicit discretization framework of the Langevin/implicit-Eu ler scheme, LI. Specifically, six schemes are constructed through diff erent discretization combinations of acceleration, velocity, and posit ion. Among them, the explicit BBK method (LE in our notation) and LI a re recovered, and the other four (all implicit) are named LIM1, LIM2, MID1, and MID2. The last two correspond, respectively, to the well-kno wn 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 n umerical damping. However, numerical results reveal limitations on oth er grounds. From simulations on a model butane, we find that the nonda mping methods give similar results when the timestep is small; however , as the timestep increases, LIM2 exhibits a pronounced rise in the po tential energy and produces wider distributions for the bond lengths. MID1 and MID2 appear to be the best among those implicit schemes for L angevin dynamics in terms of reasonably reproducing distributions for bond lengths, bond angles and dihedral angles (in comparison to 1 fs t imestep 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 demand ing. Nonetheless, we observe that for moderate timesteps, even when th e error is large for the fast motions, it is relatively small for the slow motions. This suggests that it is possible by large timestep algo rithms to capture the slow motions without resolving accurately the fa st motions.