<s>
Verlet	B-Language
integration	I-Language
(	O
)	O
is	O
a	O
numerical	O
method	O
used	O
to	O
integrate	B-Algorithm
Newton	O
's	O
equations	O
of	O
motion	O
.	O
</s>
<s>
It	O
is	O
frequently	O
used	O
to	O
calculate	O
trajectories	B-Application
of	O
particles	O
in	O
molecular	O
dynamics	O
simulations	O
and	O
computer	O
graphics	O
.	O
</s>
<s>
It	O
was	O
also	O
used	O
by	O
P	O
.	O
H	O
.	O
Cowell	O
and	O
A	O
.	O
C	O
.	O
C	O
.	O
Crommelin	O
in	O
1909	O
to	O
compute	O
the	O
orbit	O
of	O
Halley	B-Application
's	I-Application
Comet	I-Application
,	O
and	O
by	O
Carl	O
Størmer	O
in	O
1907	O
to	O
study	O
the	O
trajectories	B-Application
of	O
electrical	O
particles	O
in	O
a	O
magnetic	O
field	O
(	O
hence	O
it	O
is	O
also	O
called	O
Störmer	B-Language
's	I-Language
method	I-Language
)	O
.	O
</s>
<s>
The	O
Verlet	O
integrator	O
provides	O
good	O
numerical	B-Algorithm
stability	I-Algorithm
,	O
as	O
well	O
as	O
other	O
properties	O
that	O
are	O
important	O
in	O
physical	O
systems	O
such	O
as	O
time	O
reversibility	O
and	O
preservation	B-Algorithm
of	I-Algorithm
the	I-Algorithm
symplectic	I-Algorithm
form	I-Algorithm
on	I-Algorithm
phase	I-Algorithm
space	I-Algorithm
,	O
at	O
no	O
significant	O
additional	O
computational	O
cost	O
over	O
the	O
simple	O
Euler	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
is	O
the	O
mass	B-General_Concept
matrix	I-General_Concept
,	O
typically	O
diagonal	O
with	O
blocks	O
with	O
mass	O
for	O
every	O
particle	O
.	O
</s>
<s>
The	O
task	O
is	O
to	O
construct	O
a	O
sequence	O
of	O
points	O
that	O
closely	O
follow	O
the	O
points	O
on	O
the	O
trajectory	B-Application
of	O
the	O
exact	O
solution	O
.	O
</s>
<s>
Where	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
uses	O
the	O
forward	B-Algorithm
difference	I-Algorithm
approximation	O
to	O
the	O
first	O
derivative	O
in	O
differential	O
equations	O
of	O
order	O
one	O
,	O
Verlet	B-Language
integration	I-Language
can	O
be	O
seen	O
as	O
using	O
the	O
central	B-Algorithm
difference	I-Algorithm
approximation	I-Algorithm
to	O
the	O
second	O
derivative	O
:	O
</s>
<s>
A	O
disadvantage	O
of	O
the	O
Störmer	O
–	O
Verlet	B-Language
method	I-Language
is	O
that	O
if	O
the	O
time	O
step	O
(	O
)	O
changes	O
,	O
the	O
method	O
does	O
not	O
approximate	O
the	O
solution	O
to	O
the	O
differential	O
equation	O
.	O
</s>
<s>
This	O
deficiency	O
can	O
either	O
be	O
dealt	O
with	O
using	O
the	O
velocity	O
Verlet	O
algorithm	O
or	O
by	O
estimating	O
the	O
velocity	O
using	O
the	O
position	O
terms	O
and	O
the	O
mean	B-Algorithm
value	I-Algorithm
theorem	I-Algorithm
:	O
</s>
<s>
A	O
related	O
,	O
and	O
more	O
commonly	O
used	O
,	O
algorithm	O
is	O
the	O
velocity	O
Verlet	O
algorithm	O
,	O
similar	O
to	O
the	O
leapfrog	B-Algorithm
method	I-Algorithm
,	O
except	O
that	O
the	O
velocity	O
and	O
position	O
are	O
calculated	O
at	O
the	O
same	O
value	O
of	O
the	O
time	O
variable	O
(	O
leapfrog	O
does	O
not	O
,	O
as	O
the	O
name	O
suggests	O
)	O
.	O
</s>
<s>
This	O
algorithm	O
also	O
works	O
with	O
variable	O
time	O
steps	O
,	O
and	O
is	O
identical	O
to	O
the	O
'	O
kick-drift-kick	O
'	O
form	O
of	O
leapfrog	B-Algorithm
method	I-Algorithm
integration	O
.	O
</s>
<s>
One	O
might	O
note	O
that	O
the	O
long-term	O
results	O
of	O
velocity	O
Verlet	O
,	O
and	O
similarly	O
of	O
leapfrog	O
are	O
one	O
order	O
better	O
than	O
the	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
only	O
difference	O
is	O
that	O
the	O
midpoint	O
velocity	O
in	O
velocity	O
Verlet	O
is	O
considered	O
the	O
final	O
velocity	O
in	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
global	O
error	O
of	O
all	O
Euler	B-Algorithm
methods	I-Algorithm
is	O
of	O
order	O
one	O
,	O
whereas	O
the	O
global	O
error	O
of	O
this	O
method	O
is	O
,	O
similar	O
to	O
the	O
midpoint	B-Algorithm
method	I-Algorithm
,	O
of	O
order	O
two	O
.	O
</s>
<s>
Additionally	O
,	O
if	O
the	O
acceleration	O
indeed	O
results	O
from	O
the	O
forces	O
in	O
a	O
conservative	O
mechanical	O
or	O
Hamiltonian	O
system	O
,	O
the	O
energy	O
of	O
the	O
approximation	O
essentially	O
oscillates	O
around	O
the	O
constant	O
energy	O
of	O
the	O
exactly	O
solved	O
system	O
,	O
with	O
a	O
global	O
error	O
bound	O
again	O
of	O
order	O
one	O
for	O
semi-explicit	B-Algorithm
Euler	I-Algorithm
and	O
order	O
two	O
for	O
Verlet-leapfrog	O
.	O
</s>
<s>
The	O
same	O
goes	O
for	O
all	O
other	O
conserved	O
quantities	O
of	O
the	O
system	O
like	O
linear	O
or	O
angular	O
momentum	O
,	O
that	O
are	O
always	O
preserved	O
or	O
nearly	O
preserved	O
in	O
a	O
symplectic	B-Algorithm
integrator	I-Algorithm
.	O
</s>
<s>
The	O
velocity	O
Verlet	B-Language
method	I-Language
is	O
a	O
special	O
case	O
of	O
the	O
Newmark-beta	B-Algorithm
method	I-Algorithm
with	O
and	O
.	O
</s>
<s>
The	O
global	O
truncation	O
error	O
of	O
the	O
Verlet	B-Language
method	I-Language
is	O
,	O
both	O
for	O
position	O
and	O
velocity	O
.	O
</s>
<s>
Systems	O
of	O
multiple	O
particles	O
with	O
constraints	O
are	O
simpler	O
to	O
solve	O
with	O
Verlet	B-Language
integration	I-Language
than	O
with	O
Euler	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Verlet	B-Language
integration	I-Language
is	O
useful	O
because	O
it	O
directly	O
relates	O
the	O
force	O
to	O
the	O
position	O
,	O
rather	O
than	O
solving	O
the	O
problem	O
using	O
velocities	O
.	O
</s>
<s>
One	O
way	O
to	O
solve	O
this	O
is	O
to	O
loop	O
through	O
every	O
point	O
in	O
a	O
simulation	O
,	O
so	O
that	O
at	O
every	O
point	O
the	O
constraint	B-Algorithm
relaxation	O
of	O
the	O
last	O
is	O
already	O
used	O
to	O
speed	O
up	O
the	O
spread	O
of	O
the	O
information	O
.	O
</s>
<s>
In	O
a	O
simulation	O
this	O
may	O
be	O
implemented	O
by	O
using	O
small	O
time	O
steps	O
for	O
the	O
simulation	O
,	O
using	O
a	O
fixed	O
number	O
of	O
constraint-solving	O
steps	O
per	O
time	O
step	O
,	O
or	O
solving	O
constraints	O
until	O
they	O
are	O
met	O
by	O
a	O
specific	O
deviation	O
.	O
</s>
<s>
When	O
approximating	O
the	O
constraints	O
locally	O
to	O
first	O
order	O
,	O
this	O
is	O
the	O
same	O
as	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
For	O
small	O
matrices	B-Architecture
it	O
is	O
known	O
that	O
LU	O
decomposition	O
is	O
faster	O
.	O
</s>
<s>
Inside	O
clusters	O
the	O
LU	O
method	O
is	O
used	O
,	O
between	O
clusters	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
is	O
used	O
.	O
</s>
<s>
The	O
matrix	O
code	O
can	O
be	O
reused	O
:	O
The	O
dependency	O
of	O
the	O
forces	O
on	O
the	O
positions	O
can	O
be	O
approximated	O
locally	O
to	O
first	O
order	O
,	O
and	O
the	O
Verlet	B-Language
integration	I-Language
can	O
be	O
made	O
more	O
implicit	O
.	O
</s>
<s>
Sophisticated	O
software	O
,	O
such	O
as	O
SuperLU	O
exists	O
to	O
solve	O
complex	O
problems	O
using	O
sparse	O
matrices	B-Architecture
.	O
</s>
<s>
Specific	O
techniques	O
,	O
such	O
as	O
using	O
(	O
clusters	O
of	O
)	O
matrices	B-Architecture
,	O
may	O
be	O
used	O
to	O
address	O
the	O
specific	O
problem	O
,	O
such	O
as	O
that	O
of	O
force	O
propagating	O
through	O
a	O
sheet	O
of	O
cloth	O
without	O
forming	O
a	O
sound	O
wave	O
.	O
</s>
<s>
Another	O
way	O
to	O
solve	O
holonomic	O
constraints	O
is	O
to	O
use	O
constraint	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
The	O
Verlet	B-Language
integration	I-Language
would	O
automatically	O
handle	O
the	O
velocity	O
imparted	O
by	O
the	O
collision	O
in	O
the	O
latter	O
case	O
;	O
however	O
,	O
note	O
that	O
this	O
is	O
not	O
guaranteed	O
to	O
do	O
so	O
in	O
a	O
way	O
that	O
is	O
consistent	O
with	O
collision	O
physics	O
(	O
that	O
is	O
,	O
changes	O
in	O
momentum	O
are	O
not	O
guaranteed	O
to	O
be	O
realistic	O
)	O
.	O
</s>
