<s>
Flux	B-Algorithm
limiters	I-Algorithm
are	O
used	O
in	O
high	B-Algorithm
resolution	I-Algorithm
schemes	I-Algorithm
–	O
numerical	O
schemes	O
used	O
to	O
solve	O
problems	O
in	O
science	O
and	O
engineering	O
,	O
particularly	O
fluid	O
dynamics	O
,	O
described	O
by	O
partial	O
differential	O
equations	O
(	O
PDEs	O
)	O
.	O
</s>
<s>
They	O
are	O
used	O
in	O
high	B-Algorithm
resolution	I-Algorithm
schemes	I-Algorithm
,	O
such	O
as	O
the	O
MUSCL	O
scheme	O
,	O
to	O
avoid	O
the	O
spurious	O
oscillations	O
(	O
wiggles	O
)	O
that	O
would	O
otherwise	O
occur	O
with	O
high	O
order	O
spatial	O
discretization	O
schemes	O
due	O
to	O
shocks	O
,	O
discontinuities	O
or	O
sharp	O
changes	O
in	O
the	O
solution	O
domain	O
.	O
</s>
<s>
Use	O
of	O
flux	B-Algorithm
limiters	I-Algorithm
,	O
together	O
with	O
an	O
appropriate	O
high	B-Algorithm
resolution	I-Algorithm
scheme	I-Algorithm
,	O
make	O
the	O
solutions	O
total	B-Algorithm
variation	I-Algorithm
diminishing	I-Algorithm
(	O
TVD	B-Algorithm
)	O
.	O
</s>
<s>
Note	O
that	O
flux	B-Algorithm
limiters	I-Algorithm
are	O
also	O
referred	O
to	O
as	O
slope	B-Algorithm
limiters	I-Algorithm
because	O
they	O
both	O
have	O
the	O
same	O
mathematical	O
form	O
,	O
and	O
both	O
have	O
the	O
effect	O
of	O
limiting	O
the	O
solution	O
gradient	O
near	O
shocks	O
or	O
discontinuities	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
term	O
flux	B-Algorithm
limiter	I-Algorithm
is	O
used	O
when	O
the	O
limiter	O
acts	O
on	O
system	O
fluxes	B-General_Concept
,	O
and	O
slope	B-Algorithm
limiter	I-Algorithm
is	O
used	O
when	O
the	O
limiter	O
acts	O
on	O
system	O
states	O
(	O
like	O
pressure	O
,	O
velocity	O
etc	O
.	O
</s>
<s>
The	O
main	O
idea	O
behind	O
the	O
construction	O
of	O
flux	B-Algorithm
limiter	I-Algorithm
schemes	O
is	O
to	O
limit	O
the	O
spatial	O
derivatives	O
to	O
realistic	O
values	O
–	O
for	O
scientific	O
and	O
engineering	O
problems	O
this	O
usually	O
means	O
physically	O
realisable	O
and	O
meaningful	O
values	O
.	O
</s>
<s>
They	O
are	O
used	O
in	O
high	B-Algorithm
resolution	I-Algorithm
schemes	I-Algorithm
for	O
solving	O
problems	O
described	O
by	O
PDEs	O
and	O
only	O
come	O
into	O
operation	O
when	O
sharp	O
wave	O
fronts	O
are	O
present	O
.	O
</s>
<s>
For	O
smoothly	O
changing	O
waves	O
,	O
the	O
flux	B-Algorithm
limiters	I-Algorithm
do	O
not	O
operate	O
and	O
the	O
spatial	O
derivatives	O
can	O
be	O
represented	O
by	O
higher	O
order	O
approximations	O
without	O
introducing	O
spurious	O
oscillations	O
.	O
</s>
<s>
where	O
,	O
and	O
represent	O
edge	O
fluxes	B-General_Concept
for	O
the	O
i-th	O
cell	O
.	O
</s>
<s>
If	O
these	O
edge	O
fluxes	B-General_Concept
can	O
be	O
represented	O
by	O
low	O
and	O
high	B-Algorithm
resolution	I-Algorithm
schemes	I-Algorithm
,	O
then	O
a	O
flux	B-Algorithm
limiter	I-Algorithm
can	O
switch	O
between	O
these	O
schemes	O
depending	O
upon	O
the	O
gradients	O
close	O
to	O
the	O
particular	O
cell	O
,	O
as	O
follows	O
,	O
</s>
<s>
is	O
the	O
low	O
resolution	O
flux	B-General_Concept
,	O
</s>
<s>
is	O
the	O
high	B-Algorithm
resolution	I-Algorithm
flux	B-General_Concept
,	O
</s>
<s>
Therefore	O
,	O
when	O
the	O
limiter	O
is	O
equal	O
to	O
zero	O
(	O
sharp	O
gradient	O
,	O
opposite	O
slopes	O
or	O
zero	O
gradient	O
)	O
,	O
the	O
flux	B-General_Concept
is	O
represented	O
by	O
a	O
low	O
resolution	O
scheme	O
.	O
</s>
<s>
Similarly	O
,	O
when	O
the	O
limiter	O
is	O
equal	O
to	O
1	O
(	O
smooth	O
solution	O
)	O
,	O
it	O
is	O
represented	O
by	O
a	O
high	B-Algorithm
resolution	I-Algorithm
scheme	I-Algorithm
.	O
</s>
<s>
The	O
following	O
are	O
common	O
forms	O
of	O
flux/slope	O
limiter	O
function	O
,	O
:	O
</s>
<s>
Unless	O
indicated	O
to	O
the	O
contrary	O
,	O
the	O
above	O
limiter	O
functions	O
are	O
second	O
order	O
TVD	B-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
they	O
are	O
designed	O
such	O
that	O
they	O
pass	O
through	O
a	O
certain	O
region	O
of	O
the	O
solution	O
,	O
known	O
as	O
the	O
TVD	B-Algorithm
region	O
,	O
in	O
order	O
to	O
guarantee	O
stability	O
of	O
the	O
scheme	O
.	O
</s>
<s>
Second-order	O
,	O
TVD	B-Algorithm
limiters	O
satisfy	O
at	O
least	O
the	O
following	O
criteria	O
:	O
</s>
<s>
The	O
admissible	O
limiter	O
region	O
for	O
second-order	O
TVD	B-Algorithm
schemes	I-Algorithm
is	O
shown	O
in	O
the	O
Sweby	O
Diagram	O
opposite	O
,	O
and	O
plots	O
showing	O
limiter	O
functions	O
overlaid	O
onto	O
the	O
TVD	B-Algorithm
region	O
are	O
shown	O
below	O
.	O
</s>
