<s>
Explicit	B-Algorithm
and	I-Algorithm
implicit	I-Algorithm
methods	I-Algorithm
are	O
approaches	O
used	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
for	O
obtaining	O
numerical	B-General_Concept
approximations	I-General_Concept
to	O
the	O
solutions	O
of	O
time-dependent	O
ordinary	O
and	O
partial	O
differential	O
equations	O
,	O
as	O
is	O
required	O
in	O
computer	B-Application
simulations	I-Application
of	O
physical	O
processes	O
.	O
</s>
<s>
Explicit	B-Algorithm
methods	I-Algorithm
calculate	O
the	O
state	O
of	O
a	O
system	O
at	O
a	O
later	O
time	O
from	O
the	O
state	O
of	O
the	O
system	O
at	O
the	O
current	O
time	O
,	O
while	O
implicit	B-Algorithm
methods	I-Algorithm
find	O
a	O
solution	O
by	O
solving	O
an	O
equation	O
involving	O
both	O
the	O
current	O
state	O
of	O
the	O
system	O
and	O
the	O
later	O
one	O
.	O
</s>
<s>
Implicit	B-Algorithm
methods	I-Algorithm
require	O
an	O
extra	O
computation	O
(	O
solving	O
the	O
above	O
equation	O
)	O
,	O
and	O
they	O
can	O
be	O
much	O
harder	O
to	O
implement	O
.	O
</s>
<s>
Implicit	B-Algorithm
methods	I-Algorithm
are	O
used	O
because	O
many	O
problems	O
arising	O
in	O
practice	O
are	O
stiff	B-Algorithm
,	O
for	O
which	O
the	O
use	O
of	O
an	O
explicit	B-Algorithm
method	I-Algorithm
requires	O
impractically	O
small	O
time	O
steps	O
to	O
keep	O
the	O
error	O
in	O
the	O
result	O
bounded	O
(	O
see	O
numerical	B-Algorithm
stability	I-Algorithm
)	O
.	O
</s>
<s>
For	O
such	O
problems	O
,	O
to	O
achieve	O
given	O
accuracy	O
,	O
it	O
takes	O
much	O
less	O
computational	O
time	O
to	O
use	O
an	O
implicit	B-Algorithm
method	I-Algorithm
with	O
larger	O
time	O
steps	O
,	O
even	O
taking	O
into	O
account	O
that	O
one	O
needs	O
to	O
solve	O
an	O
equation	O
of	O
the	O
form	O
(	O
1	O
)	O
at	O
each	O
time	O
step	O
.	O
</s>
<s>
That	O
said	O
,	O
whether	O
one	O
should	O
use	O
an	O
explicit	O
or	O
implicit	B-Algorithm
method	I-Algorithm
depends	O
upon	O
the	O
problem	O
to	O
be	O
solved	O
.	O
</s>
<s>
Discretize	B-Algorithm
this	O
equation	O
using	O
the	O
simplest	O
explicit	B-Algorithm
and	I-Algorithm
implicit	I-Algorithm
methods	I-Algorithm
,	O
which	O
are	O
the	O
forward	O
Euler	O
and	O
backward	O
Euler	O
methods	O
(	O
see	O
numerical	B-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
)	O
and	O
compare	O
the	O
obtained	O
schemes	O
.	O
</s>
<s>
Then	O
one	O
uses	O
root-finding	O
algorithms	O
,	O
such	O
as	O
Newton	O
's	O
method	O
,	O
to	O
find	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
.	O
</s>
<s>
This	O
can	O
be	O
numerically	B-General_Concept
solved	O
using	O
root-finding	O
algorithms	O
,	O
such	O
as	O
Newton	O
's	O
method	O
,	O
to	O
obtain	O
.	O
</s>
<s>
Crank-Nicolson	B-Algorithm
can	O
be	O
viewed	O
as	O
a	O
form	O
of	O
more	O
general	O
IMEX	O
(	O
Implicit-Explicit	O
)	O
schemes	O
.	O
</s>
