<s>
In	O
mathematics	O
,	O
the	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
also	O
called	O
symplectic	B-Algorithm
Euler	I-Algorithm
,	O
semi-explicit	B-Algorithm
Euler	I-Algorithm
,	O
Euler	O
–	O
Cromer	O
,	O
and	O
Newton	B-Algorithm
–	I-Algorithm
Størmer	I-Algorithm
–	I-Algorithm
Verlet	I-Algorithm
(	O
NSV	O
)	O
,	O
is	O
a	O
modification	O
of	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
for	O
solving	O
Hamilton	O
's	O
equations	O
,	O
a	O
system	O
of	O
ordinary	O
differential	O
equations	O
that	O
arises	O
in	O
classical	O
mechanics	O
.	O
</s>
<s>
It	O
is	O
a	O
symplectic	B-Algorithm
integrator	I-Algorithm
and	O
hence	O
it	O
yields	O
better	O
results	O
than	O
the	O
standard	O
Euler	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
difference	O
with	O
the	O
standard	O
Euler	B-Algorithm
method	I-Algorithm
is	O
that	O
the	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
uses	O
vn+1	O
in	O
the	O
equation	O
for	O
xn+1	O
,	O
while	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
uses	O
vn	O
.	O
</s>
<s>
The	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
is	O
a	O
first-order	O
integrator	O
,	O
just	O
as	O
the	O
standard	O
Euler	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
it	O
commits	O
a	O
global	O
error	O
of	O
the	O
order	O
of	O
t	O
.	O
However	O
,	O
the	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
a	O
symplectic	B-Algorithm
integrator	I-Algorithm
,	O
unlike	O
the	O
standard	O
method	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
the	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
almost	O
conserves	O
the	O
energy	O
(	O
when	O
the	O
Hamiltonian	O
is	O
time-independent	O
)	O
.	O
</s>
<s>
Often	O
,	O
the	O
energy	B-Algorithm
increases	I-Algorithm
steadily	I-Algorithm
when	O
the	O
standard	O
Euler	B-Algorithm
method	I-Algorithm
is	O
applied	O
,	O
making	O
it	O
far	O
less	O
accurate	O
.	O
</s>
<s>
Alternating	O
between	O
the	O
two	O
variants	O
of	O
the	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
leads	O
in	O
one	O
simplification	O
to	O
the	O
Störmer-Verlet	O
integration	O
and	O
in	O
a	O
slightly	O
different	O
simplification	O
to	O
the	O
leapfrog	B-Algorithm
integration	I-Algorithm
,	O
increasing	O
both	O
the	O
order	O
of	O
the	O
error	O
and	O
the	O
order	O
of	O
preservation	O
of	O
energy	O
.	O
</s>
<s>
The	O
stability	O
region	O
of	O
the	O
semi-implicit	O
method	O
was	O
presented	O
by	O
Niiranen	O
although	O
the	O
semi-implicit	B-Algorithm
Euler	I-Algorithm
was	O
misleadingly	O
called	O
symmetric	O
Euler	O
in	O
his	O
paper	O
.	O
</s>
