<s>
Numerical	B-Algorithm
methods	I-Algorithm
for	I-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
are	O
methods	O
used	O
to	O
find	O
numerical	B-General_Concept
approximations	I-General_Concept
to	O
the	O
solutions	O
of	O
ordinary	O
differential	O
equations	O
(	O
ODEs	O
)	O
.	O
</s>
<s>
Their	O
use	O
is	O
also	O
known	O
as	O
"	O
numerical	B-Algorithm
integration	I-Algorithm
"	O
,	O
although	O
this	O
term	O
can	O
also	O
refer	O
to	O
the	O
computation	O
of	O
integrals	O
.	O
</s>
<s>
In	O
addition	O
,	O
some	O
methods	O
in	O
numerical	B-Algorithm
partial	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
convert	O
the	O
partial	O
differential	O
equation	O
into	O
an	O
ordinary	O
differential	O
equation	O
,	O
which	O
must	O
then	O
be	O
solved	O
.	O
</s>
<s>
In	O
this	O
section	O
,	O
we	O
describe	O
numerical	B-General_Concept
methods	I-General_Concept
for	O
IVPs	O
,	O
and	O
remark	O
that	O
boundary	O
value	O
problems	O
(	O
BVPs	O
)	O
require	O
a	O
different	O
set	O
of	O
tools	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
shooting	B-Algorithm
method	I-Algorithm
(	O
and	O
its	O
variants	O
)	O
or	O
global	O
methods	O
like	O
finite	B-Algorithm
differences	I-Algorithm
,	O
Galerkin	B-Algorithm
methods	I-Algorithm
,	O
or	O
collocation	B-Algorithm
methods	I-Algorithm
are	O
appropriate	O
for	O
that	O
class	O
of	O
problems	O
.	O
</s>
<s>
Numerical	B-General_Concept
methods	I-General_Concept
for	O
solving	O
first-order	O
IVPs	O
often	O
fall	O
into	O
one	O
of	O
two	O
large	O
categories	O
:	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
,	O
or	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
implicit	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
include	O
Adams-Moulton	B-Algorithm
methods	I-Algorithm
,	O
and	O
backward	B-Algorithm
differentiation	I-Algorithm
methods	I-Algorithm
(	O
BDF	O
)	O
,	O
whereas	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
include	O
diagonally	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
(	O
DIRK	O
)	O
,	O
singly	O
diagonally	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
(	O
SDIRK	O
)	O
,	O
and	O
Gauss	O
–	O
Radau	O
(	O
based	O
on	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
)	O
numerical	B-General_Concept
methods	I-General_Concept
.	O
</s>
<s>
Explicit	O
examples	O
from	O
the	O
linear	B-Algorithm
multistep	I-Algorithm
family	I-Algorithm
include	O
the	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
methods	I-Algorithm
,	O
and	O
any	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
with	O
a	O
lower	O
diagonal	O
Butcher	O
tableau	O
is	O
explicit	O
.	O
</s>
<s>
A	O
loose	O
rule	O
of	O
thumb	O
dictates	O
that	O
stiff	B-Algorithm
differential	I-Algorithm
equations	I-Algorithm
require	O
the	O
use	O
of	O
implicit	O
schemes	O
,	O
whereas	O
non-stiff	O
problems	O
can	O
be	O
solved	O
more	O
efficiently	O
with	O
explicit	O
schemes	O
.	O
</s>
<s>
The	O
so-called	O
general	B-Algorithm
linear	I-Algorithm
methods	I-Algorithm
(	O
GLMs	O
)	O
are	O
a	O
generalization	O
of	O
the	O
above	O
two	O
large	O
classes	O
of	O
methods	O
.	O
</s>
<s>
We	O
choose	O
a	O
step	O
size	O
h	O
,	O
and	O
we	O
construct	O
the	O
sequence	O
We	O
denote	O
by	O
a	O
numerical	B-General_Concept
estimate	O
of	O
the	O
exact	O
solution	O
.	O
</s>
<s>
This	O
is	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
(	O
or	O
forward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
in	O
contrast	O
with	O
the	O
backward	O
Euler	B-Algorithm
method	I-Algorithm
,	O
to	O
be	O
described	O
below	O
)	O
.	O
</s>
<s>
The	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
an	O
example	O
of	O
an	O
explicit	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
we	O
get	O
the	O
backward	O
Euler	B-Algorithm
method	I-Algorithm
:	O
</s>
<s>
The	O
backward	O
Euler	B-Algorithm
method	I-Algorithm
is	O
an	O
implicit	B-Algorithm
method	I-Algorithm
,	O
meaning	O
that	O
we	O
have	O
to	O
solve	O
an	O
equation	O
to	O
find	O
yn+1	O
.	O
</s>
<s>
It	O
costs	O
more	O
time	O
to	O
solve	O
this	O
equation	O
than	O
explicit	B-Algorithm
methods	I-Algorithm
;	O
this	O
cost	O
must	O
be	O
taken	O
into	O
consideration	O
when	O
one	O
selects	O
the	O
method	O
to	O
use	O
.	O
</s>
<s>
The	O
advantage	O
of	O
implicit	B-Algorithm
methods	I-Algorithm
such	O
as	O
(	O
)	O
is	O
that	O
they	O
are	O
usually	O
more	O
stable	O
for	O
solving	O
a	O
stiff	B-Algorithm
equation	I-Algorithm
,	O
meaning	O
that	O
a	O
larger	O
step	O
size	O
h	O
can	O
be	O
used	O
.	O
</s>
<s>
The	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
often	O
not	O
accurate	O
enough	O
.	O
</s>
<s>
This	O
yields	O
a	O
so-called	O
multistep	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Perhaps	O
the	O
simplest	O
is	O
the	O
leapfrog	B-Algorithm
method	I-Algorithm
which	O
is	O
second	O
order	O
and	O
(	O
roughly	O
speaking	O
)	O
relies	O
on	O
two	O
time	O
values	O
.	O
</s>
<s>
This	O
leads	O
to	O
the	O
family	O
of	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
,	O
named	O
after	O
Carl	O
Runge	O
and	O
Martin	O
Kutta	O
.	O
</s>
<s>
Methods	O
based	O
on	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
,	O
such	O
as	O
the	O
Bulirsch	B-Algorithm
–	I-Algorithm
Stoer	I-Algorithm
algorithm	I-Algorithm
,	O
are	O
often	O
used	O
to	O
construct	O
various	O
methods	O
of	O
different	O
orders	O
.	O
</s>
<s>
dense	O
output	O
:	O
cheap	O
numerical	B-General_Concept
approximations	I-General_Concept
for	O
the	O
whole	O
integration	O
interval	O
,	O
and	O
not	O
only	O
at	O
the	O
points	O
t0	O
,	O
t1	O
,	O
t2	O
,	O
...	O
</s>
<s>
support	O
for	O
parallel	B-Operating_System
computing	I-Operating_System
.	O
</s>
<s>
This	O
class	O
includes	O
Hermite	O
–	O
Obreschkoff	O
methods	O
and	O
Fehlberg	B-Algorithm
methods	I-Algorithm
,	O
as	O
well	O
as	O
methods	O
like	O
the	O
Parker	O
–	O
Sochacki	O
method	O
or	O
Bychkov	O
–	O
Scherbakov	O
method	O
,	O
which	O
compute	O
the	O
coefficients	O
of	O
the	O
Taylor	O
series	O
of	O
the	O
solution	O
y	O
recursively	O
.	O
</s>
<s>
In	O
particular	O
,	O
Nyström	B-Algorithm
methods	I-Algorithm
work	O
directly	O
with	O
second-order	O
equations	O
.	O
</s>
<s>
geometric	B-Algorithm
integration	I-Algorithm
methods	I-Algorithm
are	O
especially	O
designed	O
for	O
special	O
classes	O
of	O
ODEs	O
(	O
for	O
example	O
,	O
symplectic	B-Algorithm
integrators	I-Algorithm
for	O
the	O
solution	O
of	O
Hamiltonian	O
equations	O
)	O
.	O
</s>
<s>
They	O
take	O
care	O
that	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
respects	O
the	O
underlying	O
structure	O
or	O
geometry	O
of	O
these	O
classes	O
.	O
</s>
<s>
Quantized	B-Algorithm
state	I-Algorithm
systems	I-Algorithm
methods	I-Algorithm
are	O
a	O
family	O
of	O
ODE	O
integration	O
methods	O
based	O
on	O
the	O
idea	O
of	O
state	O
quantization	O
.	O
</s>
<s>
For	O
applications	O
that	O
require	O
parallel	B-Operating_System
computing	I-Operating_System
on	O
supercomputers	B-Architecture
,	O
the	O
degree	O
of	O
concurrency	O
offered	O
by	O
a	O
numerical	B-General_Concept
method	O
becomes	O
relevant	O
.	O
</s>
<s>
In	O
view	O
of	O
the	O
challenges	O
from	O
exascale	B-General_Concept
computing	I-General_Concept
systems	O
,	O
numerical	B-General_Concept
methods	I-General_Concept
for	O
initial	O
value	O
problems	O
which	O
can	O
provide	O
concurrency	O
in	O
temporal	O
direction	O
are	O
being	O
studied	O
.	O
</s>
<s>
Parareal	B-Operating_System
is	O
a	O
relatively	O
well	O
known	O
example	O
of	O
such	O
a	O
parallel-in-time	O
integration	O
method	O
,	O
but	O
early	O
ideas	O
go	O
back	O
into	O
the	O
1960s	O
.	O
</s>
<s>
In	O
the	O
advent	O
of	O
exascale	B-General_Concept
computing	I-General_Concept
,	O
time-parallel	O
integration	O
methods	O
receive	O
again	O
increased	O
attention	O
.	O
</s>
<s>
Algorithms	O
for	O
exponential	O
integrators	O
can	O
leverage	O
e.g.	O
,	O
the	O
standardized	O
Batched	O
BLAS	B-Application
functions	O
that	O
allow	O
an	O
easy	O
and	O
efficient	O
implementation	O
of	O
parallelized	B-Operating_System
integrators	O
.	O
</s>
<s>
Numerical	B-General_Concept
analysis	I-General_Concept
is	O
not	O
only	O
the	O
design	O
of	O
numerical	B-General_Concept
methods	I-General_Concept
,	O
but	O
also	O
their	O
analysis	O
.	O
</s>
<s>
A	O
numerical	B-General_Concept
method	O
is	O
said	O
to	O
be	O
convergent	O
if	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
approaches	O
the	O
exact	O
solution	O
as	O
the	O
step	O
size	O
h	O
goes	O
to	O
0	O
.	O
</s>
<s>
The	O
(	O
forward	O
)	O
Euler	B-Algorithm
method	I-Algorithm
(	O
4	O
)	O
and	O
the	O
backward	O
Euler	B-Algorithm
method	I-Algorithm
(	O
6	O
)	O
introduced	O
above	O
both	O
have	O
order	O
1	O
,	O
so	O
they	O
are	O
consistent	O
.	O
</s>
<s>
For	O
some	O
differential	O
equations	O
,	O
application	O
of	O
standard	O
methods	O
—	O
such	O
as	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
explicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
,	O
or	O
multistep	B-Algorithm
methods	I-Algorithm
(	O
for	O
example	O
,	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
methods	I-Algorithm
)	O
—	O
exhibit	O
instability	O
in	O
the	O
solutions	O
,	O
though	O
other	O
methods	O
may	O
produce	O
stable	O
solutions	O
.	O
</s>
<s>
This	O
"	O
difficult	O
behaviour	O
"	O
in	O
the	O
equation	O
(	O
which	O
may	O
not	O
necessarily	O
be	O
complex	O
itself	O
)	O
is	O
described	O
as	O
stiffness	B-Algorithm
,	O
and	O
is	O
often	O
caused	O
by	O
the	O
presence	O
of	O
different	O
time	O
scales	O
in	O
the	O
underlying	O
problem	O
.	O
</s>
<s>
Stiff	B-Algorithm
problems	I-Algorithm
are	O
ubiquitous	O
in	O
chemical	O
kinetics	O
,	O
control	O
theory	O
,	O
solid	O
mechanics	O
,	O
weather	O
forecasting	O
,	O
biology	O
,	O
plasma	O
physics	O
,	O
and	O
electronics	O
.	O
</s>
<s>
One	O
way	O
to	O
overcome	O
stiffness	B-Algorithm
is	O
to	O
extend	O
the	O
notion	O
of	O
differential	O
equation	O
to	O
that	O
of	O
differential	O
inclusion	O
,	O
which	O
allows	O
for	O
and	O
models	O
non-smoothness	O
.	O
</s>
<s>
Below	O
is	O
a	O
timeline	B-Application
of	O
some	O
important	O
developments	O
in	O
this	O
field	O
.	O
</s>
<s>
1824	O
-	O
Augustin	O
Louis	O
Cauchy	O
proves	O
convergence	O
of	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
In	O
this	O
proof	O
,	O
Cauchy	O
uses	O
the	O
implicit	O
Euler	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
1855	O
-	O
First	O
mention	O
of	O
the	O
multistep	B-Algorithm
methods	I-Algorithm
of	O
John	O
Couch	O
Adams	O
in	O
a	O
letter	O
written	O
by	O
Francis	O
Bashforth	O
.	O
</s>
<s>
1895	O
-	O
Carl	O
Runge	O
publishes	O
the	O
first	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
1901	O
-	O
Martin	O
Kutta	O
describes	O
the	O
popular	O
fourth-order	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
1910	O
-	O
Lewis	O
Fry	O
Richardson	O
announces	O
his	O
extrapolation	O
method	O
,	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
.	O
</s>
<s>
1952	O
-	O
Charles	O
F	O
.	O
Curtiss	O
and	O
Joseph	O
Oakland	O
Hirschfelder	O
coin	O
the	O
term	O
stiff	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
Boundary	O
value	O
problems	O
(	O
BVPs	O
)	O
are	O
usually	O
solved	O
numerically	B-General_Concept
by	O
solving	O
an	O
approximately	O
equivalent	O
matrix	O
problem	O
obtained	O
by	O
discretizing	O
the	O
original	O
BVP	O
.	O
</s>
<s>
The	O
most	O
commonly	O
used	O
method	O
for	O
numerically	B-General_Concept
solving	O
BVPs	O
in	O
one	O
dimension	O
is	O
called	O
the	O
Finite	B-Algorithm
Difference	I-Algorithm
Method	I-Algorithm
.	O
</s>
<s>
This	O
method	O
takes	O
advantage	O
of	O
linear	O
combinations	O
of	O
point	O
values	O
to	O
construct	O
finite	B-Algorithm
difference	I-Algorithm
coefficients	I-Algorithm
that	O
describe	O
derivatives	O
of	O
the	O
function	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
second-order	O
central	B-Algorithm
difference	I-Algorithm
approximation	I-Algorithm
to	O
the	O
first	O
derivative	O
is	O
given	O
by	O
:	O
</s>
<s>
and	O
the	O
second-order	O
central	B-Algorithm
difference	I-Algorithm
for	O
the	O
second	O
derivative	O
is	O
given	O
by	O
:	O
</s>
