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md2011-03-note_Part_7

# md2011-03-note_Part_7 - First new positions are calculated...

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Unformatted text preview: First, new positions are calculated from the ﬁrst equation. Then, from the equations of motion, the a(t + ∆t ) is calculated and then used for the new velocities v(t + ∆t ). Velocity Verlet can also be implemented in a two-step predictor-corrector algorithm with a predictor step: r(t + ∆t ) 1 = r(t ) + ∆t v(t ) + 2 ∆t 2 a(t ) (17) v p t+ 1 2 ∆t = v(t ) + 1 2 ∆t a(t ) and a corrector step: 1 vc (t + ∆t ) = vp t + ∆t 2 Notes 1 + ∆t a(t + ∆t ). 2 (18) At this point, the K is available, and the U has been evaluated when calculating the forces. Velocity Verlet doesn’t require more memory than the standard Verlet, and its numerical stability, convenience and simplicity make it a very attractive candidate for MD codes. There exists a variety of other predictor-corrector algorithms to implement. A few more can be found in Allen & Tildeslay.a a Allen (1987) Notes and Tildeslay, Computer Simulation of Liquids, Oxford University Press C V A The 3 boxes denote diﬀerent times (t − ∆t , t , t + ∆t ) and shaded boxes correspond to properties stored in memory. Notes C Notes E C S For solving the equations of motion, we need a time step ∆t . A long one means fast simulations, but causes problems such as non-conserved energy. If no energetic processes are involved, typically ∆t ≈ 1 − 10 fs is reasonable. To reduce the O(N 2 ) scalability, a neighbour list is generated. The equations of motion are solved with a predictor – corrector algorithm, for which many alternatives exist. The idea is to use Taylor expansion over the time-derivatives of r(t ) and to correct the made guesses with correct a(t ) calculated from fi (r, t ) = mi ai (t ). Notes ...
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md2011-03-note_Part_7 - First new positions are calculated...

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