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