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.
 
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