<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
local	O
linearization	O
(	O
LL	O
)	O
method	O
is	O
a	O
general	O
strategy	O
for	O
designing	O
numerical	B-Algorithm
integrators	I-Algorithm
for	O
differential	O
equations	O
based	O
on	O
a	O
local	O
(	O
piecewise	O
)	O
linearization	O
of	O
the	O
given	O
equation	O
on	O
consecutive	O
time	O
intervals	O
.	O
</s>
<s>
The	O
numerical	B-Algorithm
integrators	I-Algorithm
are	O
then	O
iteratively	O
defined	O
as	O
the	O
solution	O
of	O
the	O
resulting	O
piecewise	O
linear	O
equation	O
at	O
the	O
end	O
of	O
each	O
consecutive	O
interval	O
.	O
</s>
<s>
However	O
,	O
since	O
the	O
exact	O
solutions	O
of	O
these	O
equations	O
are	O
usually	O
unknown	O
,	O
numerical	B-General_Concept
approximations	I-General_Concept
to	O
them	O
obtained	O
by	O
numerical	B-Algorithm
integrators	I-Algorithm
are	O
necessary	O
.	O
</s>
<s>
Currently	O
,	O
many	O
applications	O
in	O
engineering	O
and	O
applied	O
sciences	O
focused	O
in	O
dynamical	O
studies	O
demand	O
the	O
developing	O
of	O
efficient	O
numerical	B-Algorithm
integrators	I-Algorithm
that	O
preserve	O
,	O
as	O
much	O
as	O
possible	O
,	O
the	O
dynamics	O
of	O
these	O
equations	O
.	O
</s>
<s>
High-order	O
local	O
linearization	O
(	O
HOLL	O
)	O
method	O
is	O
a	O
generalization	O
of	O
the	O
Local	B-Algorithm
Linearization	I-Algorithm
method	I-Algorithm
oriented	O
to	O
obtain	O
high-order	O
integrators	O
for	O
differential	O
equations	O
that	O
preserve	O
the	O
stability	O
and	O
dynamics	O
of	O
the	O
linear	O
equations	O
.	O
</s>
<s>
A	O
Local	O
Linearization	O
(	O
LL	O
)	O
scheme	O
is	O
the	O
final	O
recursive	O
algorithm	O
that	O
allows	O
the	O
numerical	O
implementation	O
of	O
a	O
discretization	B-Algorithm
derived	O
from	O
the	O
LL	O
or	O
HOLL	O
method	O
for	O
a	O
class	O
of	O
differential	O
equations	O
.	O
</s>
<s>
Let	O
be	O
a	O
time	O
discretization	B-Algorithm
of	O
the	O
time	O
interval	O
with	O
maximum	O
stepsize	O
h	O
such	O
that	O
and	O
.	O
</s>
<s>
The	O
Local	O
Linear	O
discretization	B-Algorithm
(	O
4.3	O
)	O
converges	B-Algorithm
with	O
order	O
2	O
to	O
the	O
solution	O
of	O
nonlinear	O
ODEs	O
,	O
but	O
it	O
match	O
the	O
solution	O
of	O
the	O
linear	O
ODEs	O
.	O
</s>
<s>
The	O
recursion	O
(	O
4.3	O
)	O
is	O
also	O
known	O
as	O
Exponential	O
Euler	O
discretization	B-Algorithm
.	O
</s>
<s>
where	O
is	O
an	O
order	O
(	O
>2	O
)	O
approximation	O
to	O
the	O
residual	O
r	O
The	O
HOLL	O
discretization	B-Algorithm
(	O
4.4	O
)	O
converges	B-Algorithm
with	O
order	O
to	O
the	O
solution	O
of	O
nonlinear	O
ODEs	O
,	O
but	O
it	O
match	O
the	O
solution	O
of	O
the	O
linear	O
ODEs	O
.	O
</s>
<s>
HOLL	O
discretizations	B-Algorithm
are	O
,	O
for	O
instance	O
,	O
the	O
followings	O
:	O
</s>
<s>
which	O
is	O
obtained	O
by	O
solving	O
(	O
4.5	O
)	O
via	O
a	O
s-stage	O
explicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
(	O
RK	O
)	O
scheme	O
with	O
coefficients	O
.	O
</s>
<s>
which	O
results	O
from	O
the	O
interpolation	O
of	O
in	O
(	O
4.2	O
)	O
by	O
a	O
polynomial	O
of	O
degree	O
p	O
on	O
,	O
where	O
denotes	O
the	O
j-th	O
backward	B-Algorithm
difference	I-Algorithm
of	O
.	O
</s>
<s>
"	O
Dynamic	O
properties	O
of	O
the	O
Local	B-Algorithm
Linearization	I-Algorithm
method	I-Algorithm
for	O
initial-value	O
problems	O
"	O
.	O
</s>
<s>
"	O
Locally	O
Linearized	O
Runge	B-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
of	O
Dormand	O
and	O
Prince	O
"	O
.	O
</s>
<s>
with	O
,	O
and	O
are	O
the	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
coefficients	I-Algorithm
of	I-Algorithm
Dormand	I-Algorithm
and	I-Algorithm
Prince	I-Algorithm
and	O
p	O
+	O
q	O
>	O
4	O
.	O
</s>
<s>
By	O
construction	O
,	O
the	O
LL	O
and	O
HOLL	O
discretizations	B-Algorithm
inherit	O
the	O
stability	O
and	O
dynamics	O
of	O
the	O
linear	O
ODEs	O
,	O
but	O
it	O
is	O
not	O
the	O
case	O
of	O
the	O
LL	O
schemes	O
in	O
general	O
.	O
</s>
<s>
With	O
,	O
the	O
LL	O
schemes	O
(	O
4.6	O
)	O
-(4.9 )	O
are	O
A-stable	B-Algorithm
.	O
</s>
<s>
In	O
addition	O
,	O
with	O
p	O
=	O
q	O
=	O
6	O
and	O
=	O
d	O
,	O
all	O
the	O
above	O
described	O
LL	O
schemes	O
yield	O
to	O
the	O
″exact	O
computation″	O
(	O
up	O
to	O
the	O
precision	O
of	O
the	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
)	O
of	O
linear	O
ODEs	O
on	O
the	O
current	O
personal	O
computers	O
.	O
</s>
<s>
This	O
includes	O
stiff	B-Algorithm
and	O
highly	O
oscillatory	O
linear	O
equations	O
.	O
</s>
<s>
These	O
LL	O
schemes	O
are	O
also	O
linearization	O
preserving	O
,	O
and	O
display	O
a	O
better	O
reproduction	O
of	O
the	O
stable	O
and	O
unstable	O
manifolds	O
around	O
hyperbolic	O
equilibrium	O
points	O
and	O
periodic	O
orbits	O
that	O
other	B-Algorithm
numerical	I-Algorithm
schemes	I-Algorithm
with	O
the	O
same	O
stepsize	O
.	O
</s>
<s>
The	O
Local	O
Linear	O
discretization	B-Algorithm
(	O
5.2	O
)	O
converges	B-Algorithm
to	O
the	O
solution	O
of	O
(	O
5.1	O
)	O
with	O
order	O
if	O
approximates	O
with	O
order	O
for	O
all	O
.	O
</s>
<s>
Every	O
numerical	O
implementation	O
of	O
a	O
Local	O
Linear	O
discretization	B-Algorithm
is	O
generically	O
called	O
local	O
linearization	O
scheme	O
.	O
</s>
<s>
2	O
Illustrates	O
the	O
stability	O
of	O
the	O
LL	O
scheme	O
(	O
5.3	O
)	O
and	O
of	O
that	O
of	O
an	O
explicit	O
scheme	O
of	O
similar	O
order	O
in	O
the	O
integration	O
of	O
a	O
stiff	B-Algorithm
system	I-Algorithm
of	O
DDEs	O
.	O
</s>
<s>
Every	O
numerical	O
implementation	O
of	O
the	O
local	O
linear	O
discretization	B-Algorithm
is	O
generically	O
called	O
local	O
linearization	O
scheme	O
.	O
</s>
<s>
"	O
Rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
local	O
linearization	O
schemes	O
for	O
random	O
differential	O
equations	O
"	O
.	O
</s>
<s>
The	O
convergence	B-Architecture
rate	I-Architecture
of	O
both	O
schemes	O
is	O
,	O
where	O
is	O
the	O
exponent	O
of	O
the	O
Holder	O
condition	O
of	O
.	O
</s>
<s>
and	O
its	O
approximation	O
by	O
two	O
numerical	O
schemes	O
,	O
where	O
denotes	O
a	O
fractional	O
Brownian	O
process	O
with	O
Hurst	B-Algorithm
exponent	I-Algorithm
H	O
=	O
0.45	O
.	O
</s>
<s>
The	O
strong	O
Local	O
Linear	O
discretization	B-Algorithm
converges	B-Algorithm
with	O
order	O
(=	O
1	O
,	O
1.5	O
)	O
to	O
the	O
solution	O
of	O
(	O
7.1	O
)	O
.	O
</s>
<s>
The	O
strong	O
HOLL	O
discretization	B-Algorithm
converges	B-Algorithm
with	O
order	O
to	O
the	O
solution	O
of	O
(	O
7.1	O
)	O
.	O
</s>
<s>
Every	O
numerical	O
implementation	O
of	O
a	O
strong	O
Local	O
Linear	O
discretization	B-Algorithm
of	O
any	O
order	O
is	O
generically	O
called	O
Strong	O
Local	O
Linearization	O
(	O
SLL	O
)	O
scheme	O
.	O
</s>
<s>
By	O
construction	O
,	O
the	O
strong	O
LL	O
and	O
HOLL	O
discretizations	B-Algorithm
inherit	O
the	O
stability	O
and	O
dynamics	O
of	O
the	O
linear	O
SDEs	O
,	O
but	O
it	O
is	O
not	O
the	O
case	O
of	O
the	O
strong	O
LL	O
schemes	O
in	O
general	O
.	O
</s>
<s>
LL	O
schemes	O
(	O
7.2	O
)	O
-(7.5 )	O
with	O
are	O
A-stable	B-Algorithm
,	O
including	O
stiff	B-Algorithm
and	O
highly	O
oscillatory	O
linear	O
equations	O
.	O
</s>
<s>
Moreover	O
,	O
for	O
linear	O
SDEs	O
with	O
random	O
attractors	O
,	O
these	O
schemes	O
also	O
have	O
a	O
random	O
attractor	O
that	O
converges	B-Algorithm
in	O
probability	O
to	O
the	O
exact	O
one	O
as	O
the	O
stepsize	O
decreases	O
and	O
preserve	O
the	O
ergodicity	O
of	O
these	O
equations	O
for	O
any	O
stepsize	O
.	O
</s>
<s>
The	O
weak	O
Local	O
Linear	O
discretization	B-Algorithm
converges	B-Algorithm
with	O
order	O
(=	O
1	O
,	O
2	O
)	O
to	O
the	O
solution	O
of	O
(	O
8.1	O
)	O
.	O
</s>
<s>
Every	O
numerical	O
implementation	O
of	O
the	O
Weak	O
Local	O
Linear	O
discretization	B-Algorithm
is	O
generically	O
called	O
Weak	O
Local	O
Linearization	O
(	O
WLL	O
)	O
scheme	O
.	O
</s>
<s>
"	O
Convergence	B-Architecture
rate	I-Architecture
of	O
weak	O
Local	O
Linearization	O
schemes	O
for	O
stochastic	O
differential	O
equations	O
with	O
additive	O
noise	O
"	O
.	O
</s>
<s>
"	O
Weak	O
local	O
linear	O
discretizations	B-Algorithm
for	O
stochastic	O
differential	O
equations	O
:	O
convergence	O
and	O
numerical	O
schemes	O
"	O
.	O
</s>
<s>
By	O
construction	O
,	O
the	O
weak	O
LL	O
discretizations	B-Algorithm
inherit	O
the	O
stability	O
and	O
dynamics	O
of	O
the	O
linear	O
SDEs	O
,	O
but	O
it	O
is	O
not	O
the	O
case	O
of	O
the	O
weak	O
LL	O
schemes	O
in	O
general	O
.	O
</s>
<s>
This	O
includes	O
,	O
for	O
instance	O
,	O
the	O
equations	O
of	O
coupled	O
harmonic	O
oscillators	O
driven	O
by	O
random	O
force	O
,	O
and	O
large	O
systems	O
of	O
stiff	B-Algorithm
linear	O
SDEs	O
that	O
result	O
from	O
the	O
method	O
of	O
lines	O
for	O
linear	O
stochastic	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
(	O
1963	O
)	O
introduces	O
the	O
LL	O
discretization	B-Algorithm
for	O
ODEs	O
and	O
the	O
LL	O
scheme	O
based	O
on	O
Taylor	O
expansion	O
.	O
</s>
