<s>
Exponential	B-Algorithm
integrators	I-Algorithm
are	O
a	O
class	O
of	O
numerical	B-General_Concept
methods	I-General_Concept
for	O
the	O
solution	O
of	O
ordinary	O
differential	O
equations	O
,	O
specifically	O
initial	O
value	O
problems	O
.	O
</s>
<s>
This	O
large	O
class	O
of	O
methods	O
from	O
numerical	B-General_Concept
analysis	I-General_Concept
is	O
based	O
on	O
the	O
exact	O
integration	O
of	O
the	O
linear	O
part	O
of	O
the	O
initial	O
value	O
problem	O
.	O
</s>
<s>
Because	O
the	O
linear	O
part	O
is	O
integrated	O
exactly	O
,	O
this	O
can	O
help	O
to	O
mitigate	O
the	O
stiffness	B-Algorithm
of	O
a	O
differential	O
equation	O
.	O
</s>
<s>
Exponential	B-Algorithm
integrators	I-Algorithm
can	O
be	O
constructed	O
to	O
be	O
explicit	B-Algorithm
or	I-Algorithm
implicit	I-Algorithm
for	O
numerical	B-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
or	O
serve	O
as	O
the	O
time	O
integrator	O
for	O
numerical	B-Algorithm
partial	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
Numerical	B-General_Concept
methods	I-General_Concept
require	O
a	O
discretization	B-Algorithm
of	O
equation	O
(	O
2	O
)	O
.	O
</s>
<s>
Runge-Kutta	B-Algorithm
discretizations	B-Algorithm
,	O
</s>
<s>
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
or	O
a	O
variety	O
of	O
other	O
options	O
.	O
</s>
<s>
Exponential	O
Rosenbrock	O
methods	O
were	O
shown	O
to	O
be	O
very	O
efficient	O
in	O
solving	O
large	O
systems	O
of	O
stiff	B-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
,	O
usually	O
resulted	O
from	O
spatial	O
discretization	B-Algorithm
of	O
time	O
dependent	O
(	O
parabolic	O
)	O
PDEs	O
.	O
</s>
<s>
All	O
the	O
schemes	O
presented	O
below	O
fulfill	O
the	O
stiff	O
order	O
conditions	O
and	O
thus	O
are	O
also	O
suitable	O
for	O
solving	O
stiff	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
Cox	O
and	O
Matthews	O
describe	O
a	O
fourth-order	O
method	O
exponential	O
time	O
differencing	O
(	O
ETD	O
)	O
method	O
that	O
they	O
used	O
Maple	B-Language
to	O
derive	O
.	O
</s>
<s>
If	O
implemented	O
naively	O
,	O
the	O
above	O
algorithm	O
suffers	O
from	O
numerical	O
instabilities	O
due	O
to	O
floating	B-Algorithm
point	I-Algorithm
round-off	O
errors	O
.	O
</s>
<s>
Exponential	B-Algorithm
integrators	I-Algorithm
are	O
used	O
for	O
the	O
simulation	O
of	O
stiff	O
scenarios	O
in	O
scientific	O
and	O
visual	B-Algorithm
computing	I-Algorithm
,	O
for	O
example	O
in	O
molecular	O
dynamics	O
,	O
for	O
VLSI	O
circuit	O
simulation	O
,	O
and	O
in	O
computer	O
graphics	O
.	O
</s>
<s>
They	O
are	O
also	O
applied	O
in	O
the	O
context	O
of	O
hybrid	B-Algorithm
monte	I-Algorithm
carlo	I-Algorithm
methods	O
.	O
</s>
<s>
In	O
these	O
applications	O
,	O
exponential	B-Algorithm
integrators	I-Algorithm
show	O
the	O
advantage	O
of	O
large	O
time	B-Algorithm
stepping	I-Algorithm
capability	O
and	O
high	O
accuracy	O
.	O
</s>
<s>
To	O
accelerate	O
the	O
evaluation	O
of	O
matrix	O
functions	O
in	O
such	O
complex	O
scenarios	O
,	O
exponential	B-Algorithm
integrators	I-Algorithm
are	O
often	O
combined	O
with	O
Krylov	O
subspace	O
projection	O
methods	O
.	O
</s>
