<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
finite-difference	B-Algorithm
methods	I-Algorithm
(	O
FDM	O
)	O
are	O
a	O
class	O
of	O
numerical	O
techniques	O
for	O
solving	O
differential	O
equations	O
by	O
approximating	O
derivatives	B-Algorithm
with	O
finite	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
Both	O
the	O
spatial	O
domain	O
and	O
time	O
interval	O
(	O
if	O
applicable	O
)	O
are	O
discretized	B-Algorithm
,	O
or	O
broken	O
into	O
a	O
finite	O
number	O
of	O
steps	O
,	O
and	O
the	O
value	O
of	O
the	O
solution	O
at	O
these	O
discrete	O
points	O
is	O
approximated	O
by	O
solving	O
algebraic	O
equations	O
containing	O
finite	B-Algorithm
differences	I-Algorithm
and	O
values	O
from	O
nearby	O
points	O
.	O
</s>
<s>
Finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
convert	O
ordinary	O
differential	O
equations	O
(	O
ODE	O
)	O
or	O
partial	O
differential	O
equations	O
(	O
PDE	O
)	O
,	O
which	O
may	O
be	O
nonlinear	O
,	O
into	O
a	O
system	O
of	O
linear	O
equations	O
that	O
can	O
be	O
solved	O
by	O
matrix	O
algebra	O
techniques	O
.	O
</s>
<s>
Modern	O
computers	O
can	O
perform	O
these	O
linear	B-Language
algebra	I-Language
computations	O
efficiently	O
which	O
,	O
along	O
with	O
their	O
relative	O
ease	O
of	O
implementation	O
,	O
has	O
led	O
to	O
the	O
widespread	O
use	O
of	O
FDM	O
in	O
modern	O
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
Today	O
,	O
FDM	O
are	O
one	O
of	O
the	O
most	O
common	O
approaches	O
to	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
PDE	O
,	O
along	O
with	O
finite	B-Application
element	I-Application
methods	I-Application
.	O
</s>
<s>
We	O
will	O
derive	O
an	O
approximation	O
for	O
the	O
first	B-Algorithm
derivative	I-Algorithm
of	O
the	O
function	O
f	O
by	O
first	O
truncating	O
the	O
Taylor	O
polynomial	O
plus	O
remainder	O
:	O
</s>
<s>
Assuming	O
that	O
is	O
sufficiently	O
small	O
,	O
the	O
approximation	O
of	O
the	O
first	B-Algorithm
derivative	I-Algorithm
of	O
f	O
is	O
:	O
</s>
<s>
This	O
is	O
,	O
not	O
coincidentally	O
,	O
similar	O
to	O
the	O
definition	O
of	O
derivative	B-Algorithm
,	O
which	O
is	O
given	O
as	O
:	O
</s>
<s>
The	O
two	O
sources	O
of	O
error	O
in	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
are	O
round-off	B-Algorithm
error	I-Algorithm
,	O
the	O
loss	O
of	O
precision	O
due	O
to	O
computer	O
rounding	O
of	O
decimal	O
quantities	O
,	O
and	O
truncation	B-Algorithm
error	I-Algorithm
or	O
discretization	B-Algorithm
error	I-Algorithm
,	O
the	O
difference	O
between	O
the	O
exact	O
solution	O
of	O
the	O
original	O
differential	O
equation	O
and	O
the	O
exact	O
quantity	O
assuming	O
perfect	O
arithmetic	O
(	O
that	O
is	O
,	O
assuming	O
no	O
round-off	B-Algorithm
)	O
.	O
</s>
<s>
To	O
use	O
a	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
to	O
approximate	O
the	O
solution	O
to	O
a	O
problem	O
,	O
one	O
must	O
first	O
discretize	B-Algorithm
the	O
problem	O
's	O
domain	O
.	O
</s>
<s>
This	O
means	O
that	O
finite-difference	B-Algorithm
methods	I-Algorithm
produce	O
sets	O
of	O
discrete	O
numerical	B-General_Concept
approximations	I-General_Concept
to	O
the	O
derivative	B-Algorithm
,	O
often	O
in	O
a	O
"	O
time-stepping	O
"	O
manner	O
.	O
</s>
<s>
An	O
expression	O
of	O
general	O
interest	O
is	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
of	O
a	O
method	O
.	O
</s>
<s>
Typically	O
expressed	O
using	O
Big-O	O
notation	O
,	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
refers	O
to	O
the	O
error	O
from	O
a	O
single	O
application	O
of	O
a	O
method	O
.	O
</s>
<s>
That	O
is	O
,	O
it	O
is	O
the	O
quantity	O
if	O
refers	O
to	O
the	O
exact	O
value	O
and	O
to	O
the	O
numerical	B-General_Concept
approximation	I-General_Concept
.	O
</s>
<s>
The	O
remainder	O
term	O
of	O
a	O
Taylor	O
polynomial	O
is	O
convenient	O
for	O
analyzing	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
.	O
</s>
<s>
the	O
dominant	O
term	O
of	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
can	O
be	O
discovered	O
.	O
</s>
<s>
For	O
example	O
,	O
again	O
using	O
the	O
forward-difference	O
formula	O
for	O
the	O
first	B-Algorithm
derivative	I-Algorithm
,	O
knowing	O
that	O
,	O
</s>
<s>
and	O
further	O
noting	O
that	O
the	O
quantity	O
on	O
the	O
left	O
is	O
the	O
approximation	O
from	O
the	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
and	O
that	O
the	O
quantity	O
on	O
the	O
right	O
is	O
the	O
exact	O
quantity	O
of	O
interest	O
plus	O
a	O
remainder	O
,	O
clearly	O
that	O
remainder	O
is	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
,	O
in	O
this	O
case	O
,	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
proportional	O
to	O
the	O
step	O
sizes	O
.	O
</s>
<s>
The	O
quality	O
and	O
duration	O
of	O
simulated	O
FDM	O
solution	O
depends	O
on	O
the	O
discretization	B-Algorithm
equation	O
selection	O
and	O
the	O
step	O
sizes	O
(	O
time	O
and	O
space	O
steps	O
)	O
.	O
</s>
<s>
The	O
von	B-Algorithm
Neumann	I-Algorithm
and	O
Courant-Friedrichs-Lewy	B-Algorithm
criteria	O
are	O
often	O
evaluated	O
to	O
determine	O
the	O
numerical	O
model	O
stability	O
.	O
</s>
<s>
The	O
last	O
equation	O
is	O
a	O
finite-difference	B-Algorithm
equation	I-Algorithm
,	O
and	O
solving	O
this	O
equation	O
gives	O
an	O
approximate	O
solution	O
to	O
the	O
differential	O
equation	O
.	O
</s>
<s>
One	O
way	O
to	O
numerically	B-General_Concept
solve	O
this	O
equation	O
is	O
to	O
approximate	O
all	O
the	O
derivatives	B-Algorithm
by	O
finite	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
Using	O
a	O
forward	B-Algorithm
difference	I-Algorithm
at	O
time	O
and	O
a	O
second-order	O
central	B-Algorithm
difference	I-Algorithm
for	O
the	O
space	O
derivative	B-Algorithm
at	O
position	O
(	O
FTCS	B-Algorithm
)	O
we	O
get	O
the	O
recurrence	O
equation	O
:	O
</s>
<s>
This	O
is	O
an	O
explicit	B-Algorithm
method	I-Algorithm
for	O
solving	O
the	O
one-dimensional	O
heat	O
equation	O
.	O
</s>
<s>
This	O
explicit	B-Algorithm
method	I-Algorithm
is	O
known	O
to	O
be	O
numerically	B-Algorithm
stable	I-Algorithm
and	O
convergent	B-Algorithm
whenever	O
.	O
</s>
<s>
If	O
we	O
use	O
the	O
backward	B-Algorithm
difference	I-Algorithm
at	O
time	O
and	O
a	O
second-order	O
central	B-Algorithm
difference	I-Algorithm
for	O
the	O
space	O
derivative	B-Algorithm
at	O
position	O
(	O
The	O
Backward	O
Time	O
,	O
Centered	O
Space	O
Method	O
"	O
BTCS	O
"	O
)	O
we	O
get	O
the	O
recurrence	O
equation	O
:	O
</s>
<s>
This	O
is	O
an	O
implicit	B-Algorithm
method	I-Algorithm
for	O
solving	O
the	O
one-dimensional	O
heat	O
equation	O
.	O
</s>
<s>
The	O
scheme	O
is	O
always	O
numerically	B-Algorithm
stable	I-Algorithm
and	O
convergent	B-Algorithm
but	O
usually	O
more	O
numerically	B-General_Concept
intensive	O
than	O
the	O
explicit	B-Algorithm
method	I-Algorithm
as	O
it	O
requires	O
solving	O
a	O
system	O
of	O
numerical	O
equations	O
on	O
each	O
time	O
step	O
.	O
</s>
<s>
Finally	O
if	O
we	O
use	O
the	O
central	B-Algorithm
difference	I-Algorithm
at	O
time	O
and	O
a	O
second-order	O
central	B-Algorithm
difference	I-Algorithm
for	O
the	O
space	O
derivative	B-Algorithm
at	O
position	O
(	O
"	O
CTCS	O
"	O
)	O
we	O
get	O
the	O
recurrence	O
equation	O
:	O
</s>
<s>
This	O
formula	O
is	O
known	O
as	O
the	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
scheme	O
is	O
always	O
numerically	B-Algorithm
stable	I-Algorithm
and	O
convergent	B-Algorithm
but	O
usually	O
more	O
numerically	B-General_Concept
intensive	O
as	O
it	O
requires	O
solving	O
a	O
system	O
of	O
numerical	O
equations	O
on	O
each	O
time	O
step	O
.	O
</s>
<s>
To	O
summarize	O
,	O
usually	O
the	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
scheme	I-Algorithm
is	O
the	O
most	O
accurate	O
scheme	O
for	O
small	O
time	O
steps	O
.	O
</s>
<s>
The	O
explicit	O
scheme	O
is	O
the	O
least	O
accurate	O
and	O
can	O
be	O
unstable	O
,	O
but	O
is	O
also	O
the	O
easiest	O
to	O
implement	O
and	O
the	O
least	O
numerically	B-General_Concept
intensive	O
.	O
</s>
<s>
The	O
discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
depends	O
on	O
the	O
dimension	O
.	O
</s>
<s>
For	O
an	O
equidistant	O
grid	O
one	O
gets	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
To	O
prove	O
this	O
,	O
one	O
needs	O
to	O
substitute	O
Taylor	O
Series	O
expansions	O
up	O
to	O
order	O
3	O
into	O
the	O
discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
where	O
the	O
approximation	O
is	O
evaluated	O
on	O
points	O
of	O
the	O
grid	O
,	O
and	O
the	O
stencil	B-Algorithm
is	O
assumed	O
to	O
be	O
of	O
positive	O
type	O
.	O
</s>
<s>
where	O
are	O
discretizations	B-Algorithm
of	O
the	O
continuous	O
domain	O
,	O
respectively	O
the	O
boundary	O
.	O
</s>
<s>
The	O
SBP-SAT	O
(	O
summation	O
by	O
parts	O
-	O
simultaneous	O
approximation	O
term	O
)	O
method	O
is	O
a	O
stable	O
and	O
accurate	O
technique	O
for	O
discretizing	O
and	O
imposing	O
boundary	O
conditions	O
of	O
a	O
well-posed	O
partial	O
differential	O
equation	O
using	O
high	O
order	O
finite	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
The	O
method	O
is	O
based	O
on	O
finite	B-Algorithm
differences	I-Algorithm
where	O
the	O
differentiation	O
operators	O
exhibit	O
summation-by-parts	O
properties	O
.	O
</s>
<s>
Typically	O
,	O
these	O
operators	O
consist	O
of	O
differentiation	O
matrices	O
with	O
central	B-Algorithm
difference	I-Algorithm
stencils	B-Algorithm
in	O
the	O
interior	O
with	O
carefully	O
chosen	O
one-sided	O
boundary	O
stencils	B-Algorithm
designed	O
to	O
mimic	O
integration-by-parts	O
in	O
the	O
discrete	O
setting	O
.	O
</s>
<s>
This	O
guarantees	O
stability	O
if	O
an	O
integration	O
scheme	O
with	O
a	O
stability	O
region	O
that	O
includes	O
parts	O
of	O
the	O
imaginary	O
axis	O
,	O
such	O
as	O
the	O
fourth	O
order	O
Runge-Kutta	B-Algorithm
method	I-Algorithm
,	O
is	O
used	O
.	O
</s>
<s>
This	O
makes	O
the	O
SAT	O
technique	O
an	O
attractive	O
method	O
of	O
imposing	O
boundary	O
conditions	O
for	O
higher	O
order	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
,	O
in	O
contrast	O
to	O
for	O
example	O
the	O
injection	O
method	O
,	O
which	O
typically	O
will	O
not	O
be	O
stable	O
if	O
high	O
order	O
differentiation	O
operators	O
are	O
used	O
.	O
</s>
