<s>
In	O
mathematics	O
,	O
the	O
convergence	O
condition	O
by	O
Courant	B-Algorithm
–	O
Friedrichs	O
–	O
Lewy	O
is	O
a	O
necessary	O
condition	O
for	O
convergence	O
while	O
solving	O
certain	O
partial	O
differential	O
equations	O
(	O
usually	O
hyperbolic	O
PDEs	O
)	O
numerically	B-General_Concept
.	O
</s>
<s>
It	O
arises	O
in	O
the	O
numerical	B-General_Concept
analysis	I-General_Concept
of	O
explicit	B-Algorithm
time	O
integration	O
schemes	O
,	O
when	O
these	O
are	O
used	O
for	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
the	O
time	O
step	O
must	O
be	O
less	O
than	O
a	O
certain	O
time	O
in	O
many	O
explicit	B-Algorithm
time-marching	O
computer	B-Application
simulations	I-Application
,	O
otherwise	O
the	O
simulation	O
produces	O
incorrect	O
results	O
.	O
</s>
<s>
The	O
condition	O
is	O
named	O
after	O
Richard	O
Courant	B-Algorithm
,	O
Kurt	O
Friedrichs	O
,	O
and	O
Hans	O
Lewy	O
who	O
described	O
it	O
in	O
their	O
1928	O
paper	O
.	O
</s>
<s>
The	O
principle	O
behind	O
the	O
condition	O
is	O
that	O
,	O
for	O
example	O
,	O
if	O
a	O
wave	O
is	O
moving	O
across	O
a	O
discrete	O
spatial	O
grid	O
and	O
we	O
want	O
to	O
compute	O
its	O
amplitude	B-Application
at	O
discrete	O
time	O
steps	O
of	O
equal	O
duration	O
,	O
then	O
this	O
duration	O
must	O
be	O
less	O
than	O
the	O
time	O
for	O
the	O
wave	O
to	O
travel	O
to	O
adjacent	O
grid	O
points	O
.	O
</s>
<s>
Using	O
these	O
names	O
,	O
the	O
CFL	B-Algorithm
condition	I-Algorithm
relates	O
the	O
length	O
of	O
the	O
time	O
step	O
to	O
a	O
function	O
of	O
the	O
interval	O
lengths	O
of	O
each	O
spatial	O
coordinate	O
and	O
of	O
the	O
maximum	O
speed	O
that	O
information	O
can	O
travel	O
in	O
the	O
physical	O
space	O
.	O
</s>
<s>
Operatively	O
,	O
the	O
CFL	B-Algorithm
condition	I-Algorithm
is	O
commonly	O
prescribed	O
for	O
those	O
terms	O
of	O
the	O
finite-difference	B-Algorithm
approximation	I-Algorithm
of	O
general	O
partial	O
differential	O
equations	O
that	O
model	O
the	O
advection	O
phenomenon	O
.	O
</s>
<s>
The	O
CFL	B-Algorithm
condition	I-Algorithm
then	O
has	O
the	O
following	O
form	O
:	O
</s>
<s>
where	O
the	O
dimensionless	O
number	O
is	O
called	O
the	O
Courant	B-Algorithm
number	I-Algorithm
,	O
</s>
<s>
The	O
value	O
of	O
changes	O
with	O
the	O
method	O
used	O
to	O
solve	O
the	O
discretised	O
equation	O
,	O
especially	O
depending	O
on	O
whether	O
the	O
method	O
is	O
explicit	B-Algorithm
or	I-Algorithm
implicit	I-Algorithm
.	O
</s>
<s>
If	O
an	O
explicit	B-Algorithm
(	O
time-marching	O
)	O
solver	O
is	O
used	O
then	O
typically	O
.	O
</s>
<s>
By	O
analogy	O
with	O
the	O
two-dimensional	O
case	O
,	O
the	O
general	O
CFL	B-Algorithm
condition	I-Algorithm
for	O
the	O
-dimensional	O
case	O
is	O
the	O
following	O
one	O
:	O
</s>
