University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Special methods for particle dynamics
Secondorder differential equations in which firstorder derivatives do not
appear are found so frequently in applied problems, particularly those arising
from the law of motion, that special methods have been devised for their
solution.
The idea is to go directly from the second derivatives to the function itself
without having to use the first order derivatives.
( )
()
t
F
dt
t
r
d
m
2
2
=
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View Full DocumentUniversity of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Verlet algorithm for
( )
()
t
F
dt
t
r
d
m
2
2
=
r(t+h) = r(th) + 2r(t) + h
2
F(t)/m + O(h
4
)
( )
( )
( )
)
h
(
O
dt
t
r
d
6
h
dt
t
r
d
2
h
dt
t
dr
h
)
t
(
r
)
h
t
(
r
4
3
3
3
2
2
2
+
+
+
+
=
+
( )
( )
( )
)
h
(
O
dt
t
r
d
6
h
dt
t
r
d
2
h
dt
t
dr
h
)
t
(
r
)
h
t
(
r
4
3
3
3
2
2
2
+
−
+
−
=
−
Let’s write two thirdorder Taylor expansions for r(t) at t+h and th and sum
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 Fall '11
 Zhigilei
 Derivative, Kinetic Energy, Atomistic Simulations, Leonid Zhigilei, Verlet algorithm

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