symplectic integrators
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Symplectic Integrators
Marlis Hochbruck
Heinrich-Heine Universitat Dusseldorf
Oberwolfach Seminar, November 2008
Stormer-Verlet-leapfrog method
Symplectic transformationsDefinitionsFlow of Hamiltonian problems
Symplectic integratorsSymplectic Euler methodStormer-Verlet method
Stormer-Verlet-leapfrog method
system of second order differential equations
q = f (q)
◮ two-step formulation
qn+1 − 2qn + qn−1 = h2f (qn)
◮ geometric interpretation: interpolating parabola through(tn±1, qn±1), (tn, qn) s.t. parabola fulfills the ode in tn
qn−1
qn
qn+1
tn−1 tn tn+1
One-step formulation
introduce momenta (velocities) p = q, write q = f (q) as
q = p, p = f (q)
qn−1
qn
qn+1
tn−1 tn−1/2 tn tn+1/2 tn+1
qn−1/2
qn+1/2
pn−1/2
pn+1/2
pn
One-step Stormer-Verlet
Φh : (pn, qn) 7→ (pn+1, qn+1)
pn+1/2 = pn +h
2f (qn)
(A) qn+1 = qn + hpn+1/2
pn+1 = pn+1/2 +h
2f (qn+1)
dual variant Φh : (pn−1/2, qn−1/2) 7→ (pn+1/2, qn+1/2)
qn = qn−1/2 +h
2pn−1/2
(B) pn+1/2 = pn−1/2 + hf (qn)
qn+1/2 = qn +h
2pn+1/2
Partitioned systems – symplectic Euler
general partitioned problem
p = f (p, q), q = g(p, q)
variants of symplectic Euler method
pn+1/2 = pn +h
2f (qn)
qn+1/2 = qn +h
2pn+1/2
(SE1)
or
qn+1 = qn+1/2 +h
2pn+1/2
pn+1 = pn+1/2 +h
2f (qn+1)
(SE2)
Symplectic transformations
parallelogram
P = {tξ + sη | 0 ≤ s, t ≤ 1} ⊂ R2d
in (p, q) space spanned by
ξ =
[
ξp
ξq
]
, η =
[
ηp
ηq
]
◮ d = 1: oriented area
or.area(P) = det
[
ξp ηp
ξq ηq
]
= ξpηq − ξqηp
p
q
ξ
η
Higher dimensions
◮ d > 1: sum of projections of P onto coordinate planes (pi , qi )
ω(ξ, η) :=
d∑
i=1
det
[
ξpi η
pi
ξqi η
qi
]
=
d∑
i=1
(ξpi η
qi − ξ
qi η
pi )
◮ matrix notation
ω(ξ, η) = ξT Jη, with J =
[
0 id
−id 0
]
Symplectic mappings
Definition. A linear mapping A : R2d → R
2d is called symplectic if
AT JA = J
or, equivalently, if ω(Aξ,Aη) = ω(ξ, η) for all ξ, η ∈ R2d
p
q
ξ
η A
p
q
Aξ
Aη
Nonlinear symplectic maps
Definition. A differentiable map g : U → R2d is called symplectic
if the Jacobian matrix g ′(p, q) is everywhere symplectic, i.e., if
g ′(p, q)T Jg ′(p, q) = J
or, equivalently, if
ω(
g ′(p, q)ξ, g ′(p, q)η)
= ω(ξ, η)
Remark. g symplectic =⇒ g volume preserving
Example – pendulum
Area preservation of flow of Hamiltonian systems
−1 0 1 2 3 4 5 6 7 8 9
−2
−1
1
2
A
ϕπ/2(A)
ϕπ(A)
B
ϕπ/2(B)
ϕπ(B)
ϕ3π/2(B)
Theorem of Poincare
flow of Hamilton problem p = −Hq, q = Hp
ϕt : (p0, q0) 7→ (p(t), q(t))
short:
y =
[
p
q
]
, y = J−1
[
∇pH
∇qH
]
= J−1∇yH(y), J−1 = −J
Theorem (Poincare 1899). Let H(p, q) be twice differentiable onU ⊂ R
2d . Then, for each fixed t, the flow ϕt is a symplectictransformation whenever it is defined.
Symplectic Euler method
applied to p = −Hq, q = Hp:
pn+1 = pn − hHq(pn+1, qn)
qn+1 = qn + hHp(pn+1, qn)(SE1)
orqn+1 = qn + hHp(pn, qn+1)
pn+1 = pn − hHq(pn, qn+1)(SE2)
Theorem. (de Vogelaere, 1956)The symplectic Euler method is symplectic.
Theorem. The implicit midpoint rule is symplectic.
Symplectic methods II
Hamiltonian problem p = −Hq(p, q), q = Hp(p, q)
Theorem. The Stormer-Verlet method
pn+1/2 = pn −h
2Hq(pn+1/2, qn)
qn+1 = qn +h
2
(
Hp(pn+1/2, qn) + Hp(pn+1/2, qn+1)
pn+1 = pn+1/2 −h
2Hq(pn+1/2, qn+1)
is symplectic.
Area preservation of numerical flow
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
0 2 4 6 8
−2
2
explicit Euler
symplectic Euler
implicit Euler
Runge, order 2
Verlet
midpoint rule
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