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C. Tang (唐晨宇) and Y. Wang (王延颋)
Leap Frog Algorithm
The leap frog algorithm calculates velocities in between time steps, which allows it
to be suitable for cooperating with some thermostat algorithms.
If we define:
v
t −
t
2
≡
r(t) − r(t − t)
t
v
t +
t
2
≡
r(t + t) − r(t)
t
(5.6.10)
Then we have:
r(t + t) = r(t) + v
t +
t
2
· t
v
t +
t
2
= v
t −
t
2
+ a · t
(5.6.11)
Prediction-Correction Algorithm
The prediction-correction algorithm performs the integration by using the mathematical prediction-correlation method. It is asymmetric in time, namely, the inverse
of the time in doing integration does not bring the system back to the beginning point.
Let us start with the Taylor expansion:
r(t + t) = r(t) +
∂r
∂t
t +
∂
2 r
∂t 2
t
2
2!
+
∂
3 r
∂t 3
t
3
3!
+ · · ·
(5.6.12)
We can define:
x 0 (t) ≡ r(t)
x 1 (t) ≡
∂r
∂t
t
x 2 (t) ≡
∂
2 r
∂t 2
t
2
2!
x 3 (t) ≡
∂
3 r
∂t 3
t
3
3!
(5.6.13)
Thus, we have the prediction that: