<s>
The	O
finite	B-Algorithm
volume	I-Algorithm
method	I-Algorithm
(	O
FVM	O
)	O
is	O
a	O
method	O
for	O
representing	O
and	O
evaluating	O
partial	O
differential	O
equations	O
in	O
the	O
form	O
of	O
algebraic	O
equations	O
.	O
</s>
<s>
In	O
the	O
finite	B-Algorithm
volume	I-Algorithm
method	I-Algorithm
,	O
volume	O
integrals	O
in	O
a	O
partial	O
differential	O
equation	O
that	O
contain	O
a	O
divergence	B-Application
term	O
are	O
converted	O
to	O
surface	O
integrals	O
,	O
using	O
the	O
divergence	B-Application
theorem	O
.	O
</s>
<s>
These	O
terms	O
are	O
then	O
evaluated	O
as	O
fluxes	B-General_Concept
at	O
the	O
surfaces	O
of	O
each	O
finite	B-Algorithm
volume	I-Algorithm
.	O
</s>
<s>
Because	O
the	O
flux	B-General_Concept
entering	O
a	O
given	O
volume	O
is	O
identical	O
to	O
that	O
leaving	O
the	O
adjacent	O
volume	O
,	O
these	O
methods	O
are	O
conservative	O
.	O
</s>
<s>
Another	O
advantage	O
of	O
the	O
finite	B-Algorithm
volume	I-Algorithm
method	I-Algorithm
is	O
that	O
it	O
is	O
easily	O
formulated	O
to	O
allow	O
for	O
unstructured	O
meshes	O
.	O
</s>
<s>
"	O
Finite	B-Algorithm
volume	I-Algorithm
"	O
refers	O
to	O
the	O
small	O
volume	O
surrounding	O
each	O
node	O
point	O
on	O
a	O
mesh	O
.	O
</s>
<s>
Finite	B-Algorithm
volume	I-Algorithm
methods	I-Algorithm
can	O
be	O
compared	O
and	O
contrasted	O
with	O
the	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
,	O
which	O
approximate	O
derivatives	O
using	O
nodal	O
values	O
,	O
or	O
finite	B-Application
element	I-Application
methods	I-Application
,	O
which	O
create	O
local	O
approximations	O
of	O
a	O
solution	O
using	O
local	O
data	O
,	O
and	O
construct	O
a	O
global	O
approximation	O
by	O
stitching	O
them	O
together	O
.	O
</s>
<s>
In	O
contrast	O
a	O
finite	B-Algorithm
volume	I-Algorithm
method	I-Algorithm
evaluates	O
exact	O
expressions	O
for	O
the	O
average	O
value	O
of	O
the	O
solution	O
over	O
some	O
volume	O
,	O
and	O
uses	O
this	O
data	O
to	O
construct	O
approximations	O
of	O
the	O
solution	O
within	O
cells	O
.	O
</s>
<s>
Here	O
,	O
represents	O
the	O
state	O
variable	O
and	O
represents	O
the	O
flux	B-General_Concept
or	O
flow	O
of	O
.	O
</s>
<s>
If	O
we	O
assume	O
that	O
equation	O
(	O
)	O
represents	O
a	O
flowing	O
medium	O
of	O
constant	O
area	O
,	O
we	O
can	O
sub-divide	O
the	O
spatial	O
domain	O
,	O
,	O
into	O
finite	B-Algorithm
volumes	I-Algorithm
or	O
cells	O
with	O
cell	O
centers	O
indexed	O
as	O
.	O
</s>
<s>
Now	O
,	O
since	O
in	O
one	O
dimension	O
,	O
we	O
can	O
apply	O
the	O
divergence	B-Application
theorem	O
,	O
i.e.	O
</s>
<s>
,	O
and	O
substitute	O
for	O
the	O
volume	O
integral	O
of	O
the	O
divergence	B-Application
with	O
the	O
values	O
of	O
evaluated	O
at	O
the	O
cell	O
surface	O
(	O
edges	O
and	O
)	O
of	O
the	O
finite	B-Algorithm
volume	I-Algorithm
as	O
follows	O
:	O
</s>
<s>
We	O
can	O
therefore	O
derive	O
a	O
semi-discrete	O
numerical	O
scheme	O
for	O
the	O
above	O
problem	O
with	O
cell	O
centers	O
indexed	O
as	O
,	O
and	O
with	O
cell	O
edge	O
fluxes	B-General_Concept
indexed	O
as	O
,	O
by	O
differentiating	O
(	O
)	O
with	O
respect	O
to	O
time	O
to	O
obtain	O
:	O
</s>
<s>
where	O
values	O
for	O
the	O
edge	O
fluxes	B-General_Concept
,	O
,	O
can	O
be	O
reconstructed	O
by	O
interpolation	B-Algorithm
or	O
extrapolation	O
of	O
the	O
cell	O
averages	O
.	O
</s>
<s>
Here	O
,	O
represents	O
a	O
vector	O
of	O
states	O
and	O
represents	O
the	O
corresponding	O
flux	B-General_Concept
tensor	O
.	O
</s>
<s>
Again	O
we	O
can	O
sub-divide	O
the	O
spatial	O
domain	O
into	O
finite	B-Algorithm
volumes	I-Algorithm
or	O
cells	O
.	O
</s>
<s>
Again	O
,	O
values	O
for	O
the	O
edge	O
fluxes	B-General_Concept
can	O
be	O
reconstructed	O
by	O
interpolation	B-Algorithm
or	O
extrapolation	O
of	O
the	O
cell	O
averages	O
.	O
</s>
<s>
MUSCL	O
reconstruction	O
is	O
often	O
used	O
in	O
high	B-Algorithm
resolution	I-Algorithm
schemes	I-Algorithm
where	O
shocks	O
or	O
discontinuities	O
are	O
present	O
in	O
the	O
solution	O
.	O
</s>
<s>
Finite	B-Algorithm
volume	I-Algorithm
schemes	O
are	O
conservative	O
as	O
cell	O
averages	O
change	O
through	O
the	O
edge	O
fluxes	B-General_Concept
.	O
</s>
