<s>
The	O
Lax	B-Algorithm
–	I-Algorithm
Friedrichs	I-Algorithm
method	I-Algorithm
,	O
named	O
after	O
Peter	O
Lax	O
and	O
Kurt	O
O	O
.	O
Friedrichs	O
,	O
is	O
a	O
numerical	B-General_Concept
method	O
for	O
the	O
solution	O
of	O
hyperbolic	O
partial	O
differential	O
equations	O
based	O
on	O
finite	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
The	O
method	O
can	O
be	O
described	O
as	O
the	O
FTCS	B-Algorithm
(	O
forward	O
in	O
time	O
,	O
centered	O
in	O
space	O
)	O
scheme	O
with	O
a	O
numerical	B-General_Concept
dissipation	O
term	O
of	O
1/2	O
.	O
</s>
<s>
One	O
can	O
view	O
the	O
Lax	B-Algorithm
–	I-Algorithm
Friedrichs	I-Algorithm
method	I-Algorithm
as	O
an	O
alternative	O
to	O
Godunov	O
's	O
scheme	O
,	O
where	O
one	O
avoids	O
solving	O
a	O
Riemann	O
problem	O
at	O
each	O
cell	O
interface	O
,	O
at	O
the	O
expense	O
of	O
adding	O
artificial	O
viscosity	O
.	O
</s>
<s>
Then	O
the	O
Lax	B-Algorithm
–	I-Algorithm
Friedrichs	I-Algorithm
method	I-Algorithm
for	O
solving	O
the	O
above	O
partial	O
differential	O
equation	O
is	O
given	O
by	O
:	O
</s>
<s>
It	O
can	O
,	O
however	O
be	O
used	O
as	O
a	O
building	O
block	O
for	O
building	O
high-order	O
numerical	B-General_Concept
schemes	O
for	O
solving	O
hyperbolic	O
partial	O
differential	O
equations	O
,	O
much	O
like	O
Euler	O
time	O
steps	O
can	O
be	O
used	O
as	O
a	O
building	O
block	O
for	O
creating	O
high-order	O
numerical	B-General_Concept
integrators	O
for	O
ordinary	O
differential	O
equations	O
.	O
</s>
<s>
Without	O
the	O
extra	O
terms	O
and	O
in	O
the	O
discrete	O
flux	O
,	O
,	O
one	O
ends	O
up	O
with	O
the	O
FTCS	B-Algorithm
scheme	I-Algorithm
,	O
which	O
is	O
well	O
known	O
to	O
be	O
unconditionally	O
unstable	O
for	O
hyperbolic	O
problems	O
.	O
</s>
<s>
This	O
method	O
is	O
explicit	B-Algorithm
and	O
first	B-Algorithm
order	I-Algorithm
accurate	I-Algorithm
in	I-Algorithm
time	I-Algorithm
and	O
first	B-Algorithm
order	I-Algorithm
accurate	I-Algorithm
in	I-Algorithm
space	I-Algorithm
(	O
provided	O
are	O
sufficiently-smooth	O
functions	O
.	O
</s>
<s>
Under	O
these	O
conditions	O
,	O
the	O
method	O
is	O
stable	B-Algorithm
if	O
and	O
only	O
if	O
the	O
following	O
condition	O
is	O
satisfied	O
:	O
</s>
<s>
(	O
A	O
von	B-Algorithm
Neumann	I-Algorithm
stability	I-Algorithm
analysis	I-Algorithm
can	O
show	O
the	O
necessity	O
of	O
this	O
stability	O
condition	O
.	O
)	O
</s>
<s>
The	O
Lax	B-Algorithm
–	I-Algorithm
Friedrichs	I-Algorithm
method	I-Algorithm
is	O
classified	O
as	O
having	O
second-order	O
dissipation	O
and	O
third	O
order	O
dispersion	O
.	O
</s>
