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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 twostep
predictorcorrector 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 predictorcorrector
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 nonconserved 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
timederivatives 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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 Winter '12
 Kotakoski
 Numerical Analysis, Equations, Kinetic Energy, University of Oxford

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