<s>
In	O
mathematics	O
,	O
especially	O
in	O
the	O
areas	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
called	O
numerical	B-Algorithm
partial	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
,	O
a	O
compact	B-Algorithm
stencil	I-Algorithm
is	O
a	O
type	O
of	O
stencil	B-Algorithm
that	O
uses	O
only	O
nine	O
nodes	O
for	O
its	O
discretization	B-Algorithm
method	I-Algorithm
in	O
two	O
dimensions	O
.	O
</s>
<s>
For	O
any	O
structured	O
grid	O
utilizing	O
a	O
compact	B-Algorithm
stencil	I-Algorithm
in	O
1	O
,	O
2	O
,	O
or	O
3	O
dimensions	O
the	O
maximum	O
number	O
of	O
nodes	O
is	O
3	O
,	O
9	O
,	O
or	O
27	O
respectively	O
.	O
</s>
<s>
Compact	B-Algorithm
stencils	I-Algorithm
may	O
be	O
compared	O
to	O
non-compact	B-Algorithm
stencils	I-Algorithm
.	O
</s>
<s>
Compact	B-Algorithm
stencils	I-Algorithm
are	O
currently	O
implemented	O
in	O
many	O
partial	O
differential	O
equation	O
solvers	O
,	O
including	O
several	O
in	O
the	O
topics	O
of	O
CFD	O
,	O
FEA	O
,	O
and	O
other	O
mathematical	O
solvers	O
relating	O
to	O
PDE	O
's	O
.	O
</s>
<s>
The	O
two	O
point	O
stencil	B-Algorithm
for	O
the	O
first	O
derivative	O
of	O
a	O
function	O
is	O
given	O
by	O
:	O
</s>
<s>
For	O
example	O
,	O
the	O
three	O
point	O
stencil	B-Algorithm
for	O
the	O
second	O
derivative	O
of	O
a	O
function	O
is	O
given	O
by	O
:	O
</s>
