Environmental Engineering Reference
In-Depth Information
together, giving rise to a many-body problem. There are many
computational algorithms to integrate Newton's equations of
motionnumerically,withdifferentlevelsofaccuracyandcomplexity.
The basic idea is that the integration is divided into many small
stages, each separated in time by a fixed time step. The total force
on each particle at a given time,
t
, is calculated based on the
atomistic potential. From the calculation of force we can determine
the accelerations of the particles, which are then combined with
the positions and velocities at time
t
to calculate the positions and
velocities at time
t
+
t
. In this time step, it is assumed that the
force is a constant. At time
t
+
t
, based on the new positions of all
particles, the forces on the particles are determined, leading to new
positionsand velocitiesat time
t
+
2
t
,and soon.
Here we introduce two popularly used algorithms: fourth-order
Runge-Kutta method and velocity Verlet method.
Let
r
(
t
),
v
(
t
), and
a
(
t
) denote the position, velocity, and
accelerationattime
t
,respectively.
V
[
r
(
t
)]isthepotentialenergy,
m
isthemass,and
t
isthetimestep.Toupdatethesevariablesattime
t
+
t
,thefourth-orderRunge-Kuttamethodiscarriedoutbasedon
the four-step evaluation as
q
1
=
v
(
t
)
t
,
p
1
=
a
(
t
)
t
,
(1.15)
V
r
(
t
)
+
2
v
(
t
)
q
1
p
1
2
q
2
=
+
t
,
p
2
=−
t
,
(1.16)
m
V
r
(
t
)
+
2
v
(
t
)
+
q
2
p
2
2
q
3
=
t
,
p
3
=−
t
,
(1.17)
m
V
(
r
(
t
)
+
q
3
)
m
q
4
=
(
v
(
t
)
+
p
3
)
t
,
p
4
=−
t
,
(1.18)
q
1
+
2
q
2
+
2
q
3
+
q
4
6
r
(
t
+
t
)
=
r
(
t
)
+
,
p
1
+
2
p
2
+
2
p
3
+
p
4
+
=
+
v
(
t
t
)
v
(
t
)
,
6
V
(
r
(
t
+
t
))
a
(
t
+
t
)
=−
,
(1.19)
m
where
q
n
and
p
n
are auxiliary variables used in the evaluations.