<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
adaptive	B-Algorithm
mesh	I-Algorithm
refinement	I-Algorithm
(	O
AMR	O
)	O
is	O
a	O
method	O
of	O
adapting	O
the	O
accuracy	O
of	O
a	O
solution	O
within	O
certain	O
sensitive	O
or	O
turbulent	O
regions	O
of	O
simulation	O
,	O
dynamically	O
and	O
during	O
the	O
time	O
the	O
solution	O
is	O
being	O
calculated	O
.	O
</s>
<s>
When	O
solutions	O
are	O
calculated	O
numerically	B-General_Concept
,	O
they	O
are	O
often	O
limited	O
to	O
pre-determined	O
quantified	O
grids	O
as	O
in	O
the	O
Cartesian	O
plane	O
which	O
constitute	O
the	O
computational	O
grid	O
,	O
or	O
'	O
mesh	O
 '	O
.	O
</s>
<s>
Many	O
problems	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
however	O
,	O
do	O
not	O
require	O
a	O
uniform	O
precision	O
in	O
the	O
numerical	O
grids	O
used	O
for	O
graph	O
plotting	O
or	O
computational	O
simulation	O
,	O
and	O
would	O
be	O
better	O
suited	O
if	O
specific	O
areas	O
of	O
graphs	O
which	O
needed	O
precision	O
could	O
be	O
refined	O
in	O
quantification	O
only	O
in	O
the	O
regions	O
requiring	O
the	O
added	O
precision	O
.	O
</s>
<s>
Adaptive	B-Algorithm
mesh	I-Algorithm
refinement	I-Algorithm
provides	O
such	O
a	O
dynamic	O
programming	O
environment	O
for	O
adapting	O
the	O
precision	O
of	O
the	O
numerical	B-General_Concept
computation	I-General_Concept
based	O
on	O
the	O
requirements	O
of	O
a	O
computation	O
problem	O
in	O
specific	O
areas	O
of	O
multi-dimensional	O
graphs	O
which	O
need	O
precision	O
while	O
leaving	O
the	O
other	O
regions	O
of	O
the	O
multi-dimensional	O
graphs	O
at	O
lower	O
levels	O
of	O
precision	O
and	O
resolution	O
.	O
</s>
<s>
This	O
dynamic	O
technique	O
of	O
adapting	O
computation	O
precision	O
to	O
specific	O
requirements	O
has	O
been	O
accredited	O
to	O
Marsha	O
Berger	O
,	O
Joseph	O
Oliger	O
,	O
and	O
Phillip	O
Colella	O
who	O
developed	O
an	O
algorithm	O
for	O
dynamic	O
gridding	O
called	O
local	O
adaptive	B-Algorithm
mesh	I-Algorithm
refinement	I-Algorithm
.	O
</s>
<s>
The	O
use	O
of	O
AMR	O
has	O
since	O
then	O
proved	O
of	O
broad	O
use	O
and	O
has	O
been	O
used	O
in	O
studying	O
turbulence	O
problems	O
in	O
hydrodynamics	O
as	O
well	O
as	O
in	O
the	O
study	O
of	O
large	O
scale	O
structures	O
in	O
astrophysics	O
as	O
in	O
the	O
Bolshoi	B-Algorithm
Cosmological	I-Algorithm
Simulation	I-Algorithm
.	O
</s>
<s>
In	O
a	O
series	O
of	O
papers	O
,	O
Marsha	O
Berger	O
,	O
Joseph	O
Oliger	O
,	O
and	O
Phillip	O
Colella	O
developed	O
an	O
algorithm	O
for	O
dynamic	O
gridding	O
called	O
local	O
adaptive	B-Algorithm
mesh	I-Algorithm
refinement	I-Algorithm
.	O
</s>
<s>
The	O
algorithm	O
begins	O
with	O
the	O
entire	O
computational	O
domain	B-Algorithm
covered	O
with	O
a	O
coarsely	O
resolved	O
base-level	O
regular	O
Cartesian	O
grid	O
.	O
</s>
<s>
As	O
the	O
calculation	O
progresses	O
,	O
individual	O
grid	O
cells	O
are	O
tagged	O
for	O
refinement	O
,	O
using	O
a	O
criterion	O
that	O
can	O
either	O
be	O
user-supplied	O
(	O
for	O
example	O
mass	O
per	O
cell	O
remains	O
constant	O
,	O
hence	O
higher	O
density	O
regions	O
are	O
more	O
highly	O
resolved	O
)	O
or	O
based	O
on	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
.	O
</s>
<s>
This	O
allows	O
the	O
user	O
to	O
solve	O
problems	O
that	O
are	O
completely	O
intractable	O
on	O
a	O
uniform	O
grid	O
;	O
for	O
example	O
,	O
astrophysicists	O
have	O
used	O
AMR	O
to	O
model	O
a	O
collapsing	O
giant	O
molecular	O
cloud	O
core	O
down	O
to	O
an	O
effective	O
resolution	O
of	O
131,072	O
cells	O
per	O
initial	O
cloud	O
radius	B-Protocol
,	O
corresponding	O
to	O
a	O
resolution	O
of	O
1015	O
cells	O
on	O
a	O
uniform	O
grid	O
.	O
</s>
<s>
Advanced	O
mesh	B-Algorithm
refinement	I-Algorithm
has	O
been	O
introduced	O
via	O
functionals	O
.	O
</s>
<s>
The	O
limiting	O
factor	O
to	O
the	O
resolution	O
of	O
the	O
solution	O
is	O
thus	O
the	O
grid	O
spacing	O
:	O
there	O
will	O
be	O
no	O
features	O
of	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
on	O
scales	O
smaller	O
than	O
the	O
grid-spacing	O
.	O
</s>
<s>
Adaptive	B-Algorithm
mesh	I-Algorithm
refinement	I-Algorithm
(	O
AMR	O
)	O
changes	O
the	O
spacing	O
of	O
grid	O
points	O
,	O
to	O
change	O
how	O
accurately	O
the	O
solution	O
is	O
known	O
in	O
that	O
region	O
.	O
</s>
<s>
If	O
the	O
region	O
in	O
which	O
higher	O
resolution	O
is	O
desired	O
remains	O
localized	O
over	O
the	O
course	O
of	O
the	O
computation	O
,	O
then	O
static	O
mesh	B-Algorithm
refinement	I-Algorithm
can	O
be	O
used	O
-	O
in	O
which	O
the	O
grid	O
is	O
more	O
finely	O
spaced	O
in	O
some	O
regions	O
than	O
others	O
,	O
but	O
maintains	O
its	O
shape	O
over	O
time	O
.	O
</s>
<s>
Complete	O
control	O
of	O
grid	O
resolution	O
,	O
compared	O
to	O
the	O
fixed	O
resolution	O
of	O
a	O
static	O
grid	O
approach	O
,	O
or	O
the	O
Lagrangian-based	O
adaptivity	O
of	O
smoothed	B-Application
particle	I-Application
hydrodynamics	I-Application
.	O
</s>
