<s>
In	O
computational	B-Algorithm
physics	I-Algorithm
,	O
the	O
term	O
upwind	B-Algorithm
scheme	I-Algorithm
(	O
sometimes	O
advection	O
scheme	O
)	O
typically	O
refers	O
to	O
a	O
class	O
of	O
numerical	O
discretization	B-Algorithm
methods	I-Algorithm
for	O
solving	O
hyperbolic	O
partial	O
differential	O
equations	O
,	O
in	O
which	O
so-called	O
upstream	O
variables	O
are	O
used	O
to	O
calculate	O
the	O
derivatives	O
in	O
a	O
flow	O
field	O
.	O
</s>
<s>
Historically	O
,	O
the	O
origin	O
of	O
upwind	B-Algorithm
methods	I-Algorithm
can	O
be	O
traced	O
back	O
to	O
the	O
work	O
of	O
Courant	B-Algorithm
,	O
Isaacson	O
,	O
and	O
Rees	O
who	O
proposed	O
the	O
CIR	O
method	O
.	O
</s>
<s>
If	O
the	O
finite	B-Algorithm
difference	I-Algorithm
scheme	I-Algorithm
for	O
the	O
spatial	O
derivative	O
,	O
contains	O
more	O
points	O
in	O
the	O
upwind	O
side	O
,	O
the	O
scheme	O
is	O
called	O
an	O
upwind-biased	O
or	O
simply	O
an	O
upwind	B-Algorithm
scheme	I-Algorithm
.	O
</s>
<s>
The	O
simplest	O
upwind	B-Algorithm
scheme	I-Algorithm
possible	O
is	O
the	O
first-order	O
upwind	B-Algorithm
scheme	I-Algorithm
.	O
</s>
<s>
The	O
upwind	B-Algorithm
scheme	I-Algorithm
is	O
stable	B-Algorithm
if	O
the	O
following	O
Courant	B-Algorithm
–	I-Algorithm
Friedrichs	I-Algorithm
–	I-Algorithm
Lewy	I-Algorithm
condition	I-Algorithm
(	O
CFL	O
)	O
is	O
satisfied	O
.	O
</s>
<s>
A	O
Taylor	O
series	O
analysis	O
of	O
the	O
upwind	B-Algorithm
scheme	I-Algorithm
discussed	O
above	O
will	O
show	O
that	O
it	O
is	O
first-order	O
accurate	O
in	O
space	O
and	O
time	O
.	O
</s>
<s>
Modified	O
wavenumber	O
analysis	O
shows	O
that	O
the	O
first-order	O
upwind	B-Algorithm
scheme	I-Algorithm
introduces	O
severe	O
numerical	O
diffusion/dissipation	O
in	O
the	O
solution	O
where	O
large	O
gradients	O
exist	O
due	O
to	O
necessity	O
of	O
high	O
wavenumbers	O
to	O
represent	O
sharp	O
gradients	O
.	O
</s>
<s>
The	O
spatial	O
accuracy	O
of	O
the	O
first-order	O
upwind	B-Algorithm
scheme	I-Algorithm
can	O
be	O
improved	O
by	O
including	O
3	O
data	O
points	O
instead	O
of	O
just	O
2	O
,	O
which	O
offers	O
a	O
more	O
accurate	O
finite	O
difference	O
stencil	O
for	O
the	O
approximation	O
of	O
spatial	O
derivative	O
.	O
</s>
