<s>
High-resolution	B-Algorithm
schemes	I-Algorithm
are	O
used	O
in	O
the	O
numerical	O
solution	O
of	O
partial	O
differential	O
equations	O
where	O
high	O
accuracy	O
is	O
required	O
in	O
the	O
presence	O
of	O
shocks	O
or	O
discontinuities	O
.	O
</s>
<s>
Second	O
-	O
or	O
higher-order	O
spatial	O
accuracy	O
is	O
obtained	O
in	O
smooth	O
parts	O
of	O
the	O
solution	O
.	O
</s>
<s>
The	O
number	O
of	O
mesh	O
points	O
containing	O
the	O
wave	O
is	O
small	O
compared	O
with	O
a	O
first-order	O
scheme	O
with	O
similar	O
accuracy	O
.	O
</s>
<s>
Since	O
publication	O
of	O
Godunov	O
's	O
order	B-Algorithm
barrier	O
theorem	O
,	O
which	O
proved	O
that	O
linear	O
methods	O
cannot	O
provide	O
non-oscillatory	O
solutions	O
higher	O
than	O
first	O
order	B-Algorithm
(	O
Godunov	O
1954	O
,	O
Godunov	O
1959	O
)	O
,	O
these	O
difficulties	O
have	O
attracted	O
much	O
attention	O
and	O
a	O
number	O
of	O
techniques	O
have	O
been	O
developed	O
that	O
largely	O
overcome	O
these	O
problems	O
.	O
</s>
<s>
To	O
avoid	O
spurious	O
or	O
non-physical	O
oscillations	O
where	O
shocks	O
are	O
present	O
,	O
schemes	O
that	O
exhibit	O
a	O
Total	B-Algorithm
Variation	I-Algorithm
Diminishing	I-Algorithm
(	O
TVD	O
)	O
characteristic	O
are	O
especially	O
attractive	O
.	O
</s>
<s>
Two	O
techniques	O
that	O
are	O
proving	O
to	O
be	O
particularly	O
effective	O
are	O
MUSCL	O
(	O
Monotone	O
Upstream-Centered	O
Schemes	O
for	O
Conservation	O
Laws	O
)	O
,	O
a	O
flux/slope	B-Algorithm
limiter	I-Algorithm
method	O
(	O
van	O
Leer	O
1979	O
,	O
Hirsch	O
1990	O
,	O
Tannehill	O
1997	O
,	O
Laney	O
1998	O
,	O
Toro	O
1999	O
)	O
and	O
the	O
WENO	O
(	O
Weighted	O
Essentially	O
Non-Oscillatory	O
)	O
method	O
(	O
Shu	O
1998	O
,	O
Shu	O
2009	O
)	O
.	O
</s>
<s>
Both	O
methods	O
are	O
usually	O
referred	O
to	O
as	O
high	B-Algorithm
resolution	I-Algorithm
schemes	I-Algorithm
(	O
see	O
diagram	O
)	O
.	O
</s>
<s>
MUSCL	O
methods	O
are	O
generally	O
second-order	O
accurate	O
in	O
smooth	O
regions	O
(	O
although	O
they	O
can	O
be	O
formulated	O
for	O
higher	O
orders	O
)	O
and	O
provide	O
good	O
resolution	O
,	O
monotonic	O
solutions	O
around	O
discontinuities	O
.	O
</s>
<s>
For	O
problems	O
comprising	O
both	O
shocks	O
and	O
complex	O
smooth	O
solution	O
structure	O
,	O
WENO	B-Algorithm
schemes	I-Algorithm
can	O
provide	O
higher	O
accuracy	O
than	O
second-order	O
schemes	O
along	O
with	O
good	O
resolution	O
around	O
discontinuities	O
.	O
</s>
<s>
Most	O
applications	O
tend	O
to	O
use	O
a	O
fifth	O
order	B-Algorithm
accurate	O
WENO	O
scheme	O
,	O
whilst	O
higher	O
order	B-Algorithm
schemes	O
can	O
be	O
used	O
where	O
the	O
problem	O
demands	O
improved	O
accuracy	O
in	O
smooth	O
regions	O
.	O
</s>
<s>
The	O
method	O
of	O
holistic	O
discretisation	O
systematically	O
analyses	O
subgrid	O
scale	O
dynamics	O
to	O
algebraically	O
construct	O
closures	O
for	O
numerical	O
discretisations	O
that	O
are	O
both	O
accurate	O
to	O
any	O
specified	O
order	B-Algorithm
of	O
error	O
in	O
smooth	O
regions	O
,	O
and	O
automatically	O
adapt	O
to	O
cater	O
for	O
rapid	O
grid	O
variations	O
through	O
the	O
algebraic	O
learning	O
of	O
subgrid	O
structures	O
(	O
Roberts	O
2003	O
)	O
.	O
</s>
