<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
FTCS	B-Algorithm
(	O
Forward	O
Time	O
Centered	O
Space	O
)	O
method	O
is	O
a	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
used	O
for	O
numerically	B-General_Concept
solving	O
the	O
heat	O
equation	O
and	O
similar	O
parabolic	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
It	O
is	O
a	O
first-order	O
method	O
in	O
time	O
,	O
explicit	B-Algorithm
in	O
time	O
,	O
and	O
is	O
conditionally	B-Algorithm
stable	I-Algorithm
when	O
applied	O
to	O
the	O
heat	O
equation	O
.	O
</s>
<s>
The	O
abbreviation	O
FTCS	B-Algorithm
was	O
first	O
used	O
by	O
Patrick	O
Roache	O
.	O
</s>
<s>
The	O
FTCS	B-Algorithm
method	I-Algorithm
is	O
based	O
on	O
the	O
forward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
in	O
time	O
(	O
hence	O
"	O
forward	O
time	O
"	O
)	O
and	O
central	B-Algorithm
difference	I-Algorithm
in	O
space	O
(	O
hence	O
"	O
centered	O
space	O
"	O
)	O
,	O
giving	O
first-order	O
convergence	O
in	O
time	O
and	O
second-order	O
convergence	O
in	O
space	O
.	O
</s>
<s>
then	O
,	O
letting	O
,	O
the	O
forward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
given	O
by	O
:	O
</s>
<s>
The	O
function	O
must	O
be	O
discretized	O
spatially	O
with	O
a	O
central	B-Algorithm
difference	I-Algorithm
scheme	O
.	O
</s>
<s>
This	O
is	O
an	O
explicit	B-Algorithm
method	I-Algorithm
which	O
means	O
that	O
,	O
can	O
be	O
explicitly	O
computed	O
(	O
no	O
need	O
of	O
solving	O
a	O
system	O
of	O
algebraic	O
equations	O
)	O
if	O
values	O
of	O
at	O
previous	O
time	O
level	O
are	O
known	O
.	O
</s>
<s>
FTCS	B-Algorithm
method	I-Algorithm
is	O
computationally	O
inexpensive	O
since	O
the	O
method	O
is	O
explicit	B-Algorithm
.	O
</s>
<s>
The	O
FTCS	B-Algorithm
method	I-Algorithm
is	O
often	O
applied	O
to	O
diffusion	O
problems	O
.	O
</s>
<s>
the	O
FTCS	B-Algorithm
scheme	I-Algorithm
is	O
given	O
by	O
:	O
</s>
<s>
As	O
derived	O
using	O
von	B-Algorithm
Neumann	I-Algorithm
stability	I-Algorithm
analysis	I-Algorithm
,	O
the	O
FTCS	B-Algorithm
method	I-Algorithm
for	O
the	O
one-dimensional	O
heat	O
equation	O
is	O
numerically	B-Algorithm
stable	I-Algorithm
if	O
and	O
only	O
if	O
the	O
following	O
condition	O
is	O
satisfied	O
:	O
</s>
<s>
Which	O
is	O
to	O
say	O
that	O
the	O
choice	O
of	O
and	O
must	O
satisfy	O
the	O
above	O
condition	O
for	O
the	O
FTCS	B-Algorithm
scheme	I-Algorithm
to	O
be	O
stable	O
.	O
</s>
<s>
A	O
major	O
drawback	O
of	O
the	O
FTCS	B-Algorithm
method	I-Algorithm
is	O
that	O
for	O
problems	O
with	O
large	O
diffusivity	O
,	O
satisfactory	O
step	O
sizes	O
can	O
be	O
too	O
small	O
to	O
be	O
practical	O
.	O
</s>
