<s>
In	O
numerical	O
mathematics	O
,	O
Beam	B-Algorithm
and	I-Algorithm
Warming	I-Algorithm
scheme	I-Algorithm
or	O
Beam	O
–	O
Warming	O
implicit	B-Algorithm
scheme	I-Algorithm
introduced	O
in	O
1978	O
by	O
Richard	O
M	O
.	O
Beam	O
and	O
R	O
.	O
F	O
.	O
Warming	O
,	O
is	O
a	O
second	O
order	O
accurate	O
implicit	B-Algorithm
scheme	I-Algorithm
,	O
mainly	O
used	O
for	O
solving	O
non-linear	O
hyperbolic	O
equations	O
.	O
</s>
<s>
This	O
scheme	O
is	O
a	O
spatially	O
factored	O
,	O
non	O
iterative	O
,	O
ADI	B-Algorithm
scheme	O
and	O
uses	O
implicit	B-Algorithm
Euler	I-Algorithm
to	O
perform	O
the	O
time	O
Integration	O
.	O
</s>
<s>
This	O
is	O
also	O
known	O
as	O
the	O
trapezoidal	B-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
This	O
resulted	O
system	O
of	O
linear	O
equations	O
can	O
be	O
solved	O
using	O
the	O
modified	O
tridiagonal	B-Language
matrix	I-Language
algorithm	I-Language
,	O
also	O
known	O
as	O
the	O
Thomas	B-Language
algorithm	I-Language
.	O
</s>
<s>
If	O
only	O
the	O
stable	O
solution	O
is	O
required	O
,	O
then	O
in	O
the	O
equation	O
to	O
the	O
right	O
hand	O
side	O
a	O
second-order	O
smoothing	B-Algorithm
term	I-Algorithm
is	O
added	O
on	O
the	O
implicit	O
layer	O
.	O
</s>
<s>
The	O
addition	O
of	O
smoothing	B-Algorithm
term	I-Algorithm
increases	O
the	O
number	O
of	O
steps	O
required	O
by	O
three	O
.	O
</s>
<s>
This	O
scheme	O
is	O
produced	O
by	O
combining	O
the	O
trapezoidal	B-Algorithm
formula	I-Algorithm
,	O
linearization	O
,	O
factoring	O
,	O
Padt	O
spatial	O
differencing	O
,	O
the	O
homogeneous	O
property	O
of	O
the	O
flux	O
vectors	O
(	O
where	O
applicable	O
)	O
,	O
and	O
hybrid	O
spatial	O
differencing	O
and	O
is	O
most	O
suitable	O
for	O
nonlinear	O
systems	O
in	O
conservation-law	O
form	O
.	O
</s>
<s>
ADI	B-Algorithm
algorithm	O
retains	O
the	O
order	O
of	O
accuracy	O
and	O
the	O
steady-state	O
property	O
while	O
reducing	O
the	O
bandwidth	O
of	O
the	O
system	O
of	O
equations	O
.	O
</s>
