<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
von	B-Algorithm
Neumann	I-Algorithm
stability	I-Algorithm
analysis	I-Algorithm
(	O
also	O
known	O
as	O
Fourier	O
stability	B-Algorithm
analysis	O
)	O
is	O
a	O
procedure	O
used	O
to	O
check	O
the	O
stability	B-Algorithm
of	O
finite	B-Algorithm
difference	I-Algorithm
schemes	I-Algorithm
as	O
applied	O
to	O
linear	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
The	O
analysis	O
is	O
based	O
on	O
the	O
Fourier	O
decomposition	O
of	O
numerical	B-Algorithm
error	I-Algorithm
and	O
was	O
developed	O
at	O
Los	O
Alamos	O
National	O
Laboratory	O
after	O
having	O
been	O
briefly	O
described	O
in	O
a	O
1947	O
article	O
by	O
British	O
researchers	O
Crank	O
and	O
Nicolson	O
.	O
</s>
<s>
The	O
stability	B-Algorithm
of	I-Algorithm
numerical	I-Algorithm
schemes	I-Algorithm
is	O
closely	O
associated	O
with	O
numerical	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
A	O
finite	B-Algorithm
difference	I-Algorithm
scheme	I-Algorithm
is	O
stable	O
if	O
the	O
errors	O
made	O
at	O
one	O
time	O
step	O
of	O
the	O
calculation	O
do	O
not	O
cause	O
the	O
errors	O
to	O
be	O
magnified	O
as	O
the	O
computations	O
are	O
continued	O
.	O
</s>
<s>
The	O
stability	B-Algorithm
of	I-Algorithm
numerical	I-Algorithm
schemes	I-Algorithm
can	O
be	O
investigated	O
by	O
performing	O
von	B-Algorithm
Neumann	I-Algorithm
stability	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
For	O
time-dependent	O
problems	O
,	O
stability	B-Algorithm
guarantees	O
that	O
the	O
numerical	O
method	O
produces	O
a	O
bounded	O
solution	O
whenever	O
the	O
solution	O
of	O
the	O
exact	O
differential	O
equation	O
is	O
bounded	O
.	O
</s>
<s>
Stability	B-Algorithm
,	O
in	O
general	O
,	O
can	O
be	O
difficult	O
to	O
investigate	O
,	O
especially	O
when	O
the	O
equation	O
under	O
consideration	O
is	O
nonlinear	O
.	O
</s>
<s>
In	O
certain	O
cases	O
,	O
von	O
Neumann	O
stability	B-Algorithm
is	O
necessary	O
and	O
sufficient	O
for	O
stability	B-Algorithm
in	O
the	O
sense	O
of	O
Lax	O
–	O
Richtmyer	O
(	O
as	O
used	O
in	O
the	O
Lax	O
equivalence	O
theorem	O
)	O
:	O
The	O
PDE	O
and	O
the	O
finite	B-Algorithm
difference	I-Algorithm
scheme	I-Algorithm
models	O
are	O
linear	O
;	O
the	O
PDE	O
is	O
constant-coefficient	O
with	O
periodic	B-Algorithm
boundary	I-Algorithm
conditions	I-Algorithm
and	O
has	O
only	O
two	O
independent	O
variables	O
;	O
and	O
the	O
scheme	O
uses	O
no	O
more	O
than	O
two	O
time	O
levels	O
.	O
</s>
<s>
Von	O
Neumann	O
stability	B-Algorithm
is	O
necessary	O
in	O
a	O
much	O
wider	O
variety	O
of	O
cases	O
.	O
</s>
<s>
It	O
is	O
often	O
used	O
in	O
place	O
of	O
a	O
more	O
detailed	O
stability	B-Algorithm
analysis	O
to	O
provide	O
a	O
good	O
guess	O
at	O
the	O
restrictions	O
(	O
if	O
any	O
)	O
on	O
the	O
step	O
sizes	O
used	O
in	O
the	O
scheme	O
because	O
of	O
its	O
relative	O
simplicity	O
.	O
</s>
<s>
where	O
is	O
the	O
solution	O
of	O
the	O
discretized	O
equation	O
(	O
)	O
that	O
would	O
be	O
computed	O
in	O
the	O
absence	O
of	O
round-off	B-Algorithm
error	I-Algorithm
,	O
and	O
is	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
obtained	O
in	O
finite	B-Algorithm
precision	I-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
Equations	O
(	O
)	O
and	O
(	O
)	O
show	O
that	O
both	O
the	O
error	O
and	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
have	O
the	O
same	O
growth	O
or	O
decay	O
behavior	O
with	O
respect	O
to	O
time	O
.	O
</s>
<s>
Often	O
the	O
assumption	O
is	O
made	O
that	O
the	O
error	O
grows	O
or	O
decays	O
exponentially	O
with	O
time	O
,	O
but	O
this	O
is	O
not	O
necessary	O
for	O
the	O
stability	B-Algorithm
analysis	O
.	O
</s>
<s>
The	O
stability	B-Algorithm
characteristics	O
can	O
be	O
studied	O
using	O
just	O
this	O
form	O
for	O
the	O
error	O
with	O
no	O
loss	O
in	O
generality	O
.	O
</s>
<s>
Equation	O
(	O
)	O
gives	O
the	O
stability	B-Algorithm
requirement	O
for	O
the	O
FTCS	B-Algorithm
scheme	I-Algorithm
as	O
applied	O
to	O
one-dimensional	O
heat	O
equation	O
.	O
</s>
<s>
Similar	O
analysis	O
shows	O
that	O
a	O
FTCS	B-Algorithm
scheme	I-Algorithm
for	O
linear	O
advection	O
is	O
unconditionally	O
unstable	O
.	O
</s>
