- Source: Geometric integrator
In the mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation.
Pendulum example
We can motivate the study of geometric integrators by considering the motion of a pendulum.
Assume that we have a pendulum whose bob has mass
m
=
1
{\displaystyle m=1}
and
whose rod is massless of length
ℓ
=
1
{\displaystyle \ell =1}
. Take the
acceleration due to gravity to be
g
=
1
{\displaystyle g=1}
. Denote by
q
(
t
)
{\displaystyle q(t)}
the angular displacement of the rod from the vertical,
and by
p
(
t
)
{\displaystyle p(t)}
the pendulum's momentum. The Hamiltonian of
the system, the sum of its kinetic and potential energies, is
H
(
q
,
p
)
=
T
(
p
)
+
U
(
q
)
=
1
2
p
2
−
cos
q
,
{\displaystyle H(q,p)=T(p)+U(q)={\frac {1}{2}}p^{2}-\cos q,}
which gives Hamilton's equations
(
q
˙
,
p
˙
)
=
(
∂
H
/
∂
p
,
−
∂
H
/
∂
q
)
=
(
p
,
−
sin
q
)
.
{\displaystyle ({\dot {q}},{\dot {p}})=(\partial H/\partial p,-\partial H/\partial q)=(p,-\sin q).\,}
It is natural to take the configuration space
Q
{\displaystyle Q}
of all
q
{\displaystyle q}
to be the unit
circle
S
1
{\displaystyle \mathbb {S} ^{1}}
, so that
(
q
,
p
)
{\displaystyle (q,p)}
lies on the
cylinder
S
1
×
R
{\displaystyle \mathbb {S} ^{1}\times \mathbb {R} }
. However, we will take
(
q
,
p
)
∈
R
2
{\displaystyle (q,p)\in \mathbb {R} ^{2}}
, simply because
(
q
,
p
)
{\displaystyle (q,p)}
-space is
then easier to plot. Define
z
(
t
)
=
(
q
(
t
)
,
p
(
t
)
)
T
{\displaystyle z(t)=(q(t),p(t))^{\mathrm {T} }}
and
f
(
z
)
=
(
p
,
−
sin
q
)
T
{\displaystyle f(z)=(p,-\sin q)^{\mathrm {T} }}
. Let us experiment by
using some simple numerical methods to integrate this system. As usual,
we select a constant step size,
h
{\displaystyle h}
, and for an arbitrary non-negative integer
k
{\displaystyle k}
we write
z
k
:=
z
(
k
h
)
{\displaystyle z_{k}:=z(kh)}
.
We use the following methods.
z
k
+
1
=
z
k
+
h
f
(
z
k
)
{\displaystyle z_{k+1}=z_{k}+hf(z_{k})\,}
(explicit Euler),
z
k
+
1
=
z
k
+
h
f
(
z
k
+
1
)
{\displaystyle z_{k+1}=z_{k}+hf(z_{k+1})\,}
(implicit Euler),
z
k
+
1
=
z
k
+
h
f
(
q
k
,
p
k
+
1
)
{\displaystyle z_{k+1}=z_{k}+hf(q_{k},p_{k+1})\,}
(symplectic Euler),
z
k
+
1
=
z
k
+
h
f
(
(
z
k
+
1
+
z
k
)
/
2
)
{\displaystyle z_{k+1}=z_{k}+hf((z_{k+1}+z_{k})/2)\,}
(implicit midpoint rule).
(Note that the symplectic Euler method treats q by the explicit and
p
{\displaystyle p}
by the implicit Euler method.)
The observation that
H
{\displaystyle H}
is constant along the solution
curves of the Hamilton's equations allows us to describe the exact
trajectories of the system: they are the level curves of
p
2
/
2
−
cos
q
{\displaystyle p^{2}/2-\cos q}
. We plot, in
R
2
{\displaystyle \mathbb {R} ^{2}}
, the exact
trajectories and the numerical solutions of the system. For the explicit
and implicit Euler methods we take
h
=
0.2
{\displaystyle h=0.2}
, and z0 = (0.5, 0) and (1.5, 0) respectively; for the other two methods we take
h
=
0.3
{\displaystyle h=0.3}
, and z0 = (0, 0.7), (0, 1.4) and (0, 2.1).
The explicit (resp. implicit) Euler method spirals out from (resp. in to) the origin. The other two methods show the correct qualitative behaviour, with the implicit midpoint rule agreeing with the exact solution to a greater degree than the symplectic Euler method.
Recall that the exact flow
ϕ
t
{\displaystyle \phi _{t}}
of a Hamiltonian system with one degree of freedom is
area-preserving, in the sense that
det
∂
ϕ
t
∂
(
q
0
,
p
0
)
=
1
{\displaystyle \det {\frac {\partial \phi _{t}}{\partial (q_{0},p_{0})}}=1}
for all
t
{\displaystyle t}
.
This formula is easily verified by hand. For our pendulum
example we see that the numerical flow
Φ
e
E
,
h
:
z
k
↦
z
k
+
1
{\displaystyle \Phi _{{\mathrm {eE} },h}:z_{k}\mapsto z_{k+1}}
of the explicit Euler method is not area-preserving; viz.,
det
∂
∂
(
q
0
,
p
0
)
Φ
e
E
,
h
(
z
0
)
=
|
1
h
−
h
cos
q
0
1
|
=
1
+
h
2
cos
q
0
.
{\displaystyle \det {\frac {\partial }{\partial (q_{0},p_{0})}}\Phi _{{\mathrm {eE} },h}(z_{0})={\begin{vmatrix}1&h\\-h\cos q_{0}&1\end{vmatrix}}=1+h^{2}\cos q_{0}.}
A similar calculation can be carried out for the implicit Euler method,
where the determinant is
det
∂
∂
(
q
0
,
p
0
)
Φ
i
E
,
h
(
z
0
)
=
(
1
+
h
2
cos
q
1
)
−
1
.
{\displaystyle \det {\frac {\partial }{\partial (q_{0},p_{0})}}\Phi _{{\mathrm {iE} },h}(z_{0})=(1+h^{2}\cos q_{1})^{-1}.}
However, the symplectic Euler method is area-preserving:
(
1
−
h
0
1
)
∂
∂
(
q
0
,
p
0
)
Φ
s
E
,
h
(
z
0
)
=
(
1
0
−
h
cos
q
0
1
)
,
{\displaystyle {\begin{pmatrix}1&-h\\0&1\end{pmatrix}}{\frac {\partial }{\partial (q_{0},p_{0})}}\Phi _{{\mathrm {sE} },h}(z_{0})={\begin{pmatrix}1&0\\-h\cos q_{0}&1\end{pmatrix}},}
thus
det
(
∂
Φ
s
E
,
h
/
∂
(
q
0
,
p
0
)
)
=
1
{\displaystyle \det(\partial \Phi _{{\mathrm {sE} },h}/\partial (q_{0},p_{0}))=1}
. The implicit midpoint rule has similar geometric properties.
To summarize: the pendulum example shows that, besides the explicit and
implicit Euler methods not being good choices of method to solve the
problem, the symplectic Euler method and implicit midpoint rule agree
well with the exact flow of the system, with the midpoint rule agreeing
more closely. Furthermore, these latter two methods are area-preserving,
just as the exact flow is; they are two examples of geometric (in fact, symplectic) integrators.
Moving frame method
The moving frame method can be used to construct numerical methods which preserve Lie symmetries of the ODE. Existing methods such as Runge-Kutta can be modified using moving frame method to produce invariant versions.
See also
Energy drift
Mimesis (mathematics)
References
Further reading
Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2002). Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer-Verlag. ISBN 3-540-43003-2.
Leimkuhler, Ben; Reich, Sebastian (2005). Simulating Hamiltonian Dynamics. Cambridge University Press. ISBN 0-521-77290-7.
Budd, C.J.; Piggott, M.D. (2003). "Geometric Integration and its Applications". Handbook of Numerical Analysis. Vol. 11. Elsevier. pp. 35–139. doi:10.1016/S1570-8659(02)11002-7. ISBN 9780444512475.
Kim, Pilwon (2007). "Invariantization of Numerical Schemes Using Moving Frames". BIT Numerical Mathematics. Vol. 47, no. 3. Springer. pp. 525–546. doi:10.1007/s10543-007-0138-8.
Kata Kunci Pencarian:
- Geometric integration
- Geometric integrator
- Symplectic integrator
- Geometric algebra
- Geometry
- Geometric series
- Lebesgue integral
- Fractional-order integrator
- Geometric calculus
- Geometric progression
