Geometric numerical integration Robert McLachlan Professor of - - PowerPoint PPT Presentation

Geometric numerical integration

Robert McLachlan Professor of Applied Mathematics, Massey University

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What is geometric integration?

A numerical method for a differential equation which inherits some property of the equation exactly.

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What is geometric integration?

A numerical method for a differential equation which inherits some property of the equation exactly. The property should be possible to impose exactly, while still constraining the solution in some useful way.

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What is geometric integration?

A numerical method for a differential equation which inherits some property of the equation exactly. The property should be possible to impose exactly, while still constraining the solution in some useful way. Examples: preserving first integrals, symmetries, phase space volume, symplecticity for Hamiltonian systems, reversibility, Lyapunov functions

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A basic example

Earth (initial position) Earth (after half a day) Earth (after a whole day)

Sun

The leapfrog method Loup Verlet, 1967

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Feynman’s Lectures on Physics

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Newton’s Principia, Book I, Theorem I

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The leapfrog method...

is simple is fast is relatively accurate is time-reversible preserves momentum and angular momentum has no drift of energy preserves quasiperiodic orbits [KAM tori] produces qualitatively correct chaotic orbits is symplectic

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Channel & Scovel, 1990

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What is symplecticity?

Phase space T ∗Q carries a symplectic 2-form ω Flow of Hamilton’s equations iXω = −dH satisfies exp(tX)∗ω = ω The leapfrog method also satisfies ϕ∗ω = ω

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What is symplecticity?

position impulse coast velocity

A C B

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Missed opportunities

Freeman Dyson “I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce.”

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The ice ages explained

(Sverker Edvarddson 2002)

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Simulation of cold water

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Simulation of cold water

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Backward error analysis

Let G be a set of diffeomorphisms with a tangent space G at the

• identity. Let ϕ = 1 + (∆t)X + o(∆t) ∈ G. Then

If ϕ ∈ Cr then there exists X ∈ G such that ϕ = exp(∆t X) + O((∆t)r) If ϕ is analytic then ϕ = exp(∆t X) + O(e−a/∆t). Therefore properties generic to G are (almost) preserved. For example, let G = symplectic maps, G=Hamiltonian vector

• fields. Then symplectic integrators have bounded energy errors

for exponentially long times O(ea/∆t).

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Splitting methods

Let X = n

i=1 Xi with X, Xi ∈ G. Use the integrators

ϕ1(∆t) = exp(∆tX1) ◦ . . . exp(∆tXn) ϕ2(∆t) = ϕ1(∆t/2)ϕ−1

1 (−∆t/2)

ϕ4(∆t) = ϕ2(z∆t)ϕ2((1 − 2z)∆t)ϕ2(z∆t), z = (2 − 21/3)−1 where ϕj = exp

• ∆tX + O((∆t)j+1)
• .

Requires knowing an explicit form (generating function) for all X ∈ G a way of constructing lots of integrable Xi ∈ G. the Baker-Campbell-Hausdorff formula exp(A) exp(B) = exp

• A + B + 1

2[A, B] + 1 6[A, [A, B]] + . . .

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Analysis of splitting methods

Quickly leads to studying L(X1, X2, . . .), the free Lie algebra generated by the Xi. Simplest and most common case: split the Hamiltonian H = T + V = 1

2pM(q)p + V (q) and study L(T, V ).

This Lie algebra is not free, because T is quadratic in p, leading to [V, [V, [V, T]]] ≡ 0. It is LP(T, V ), the Lie algebra of classical mechanics.

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Polynomial gradings

• Definition. A Lie algebra L is of class P (‘polynomially graded’) if

it is graded, i.e. LP =

n≥0 Ln, and its homogeneous

subspaces Ln satisfy [Ln, Lm] ⊆ Ln+m−1 if n > 0 or m > 0; and [L0, L0] = 0 We call the grading of L its grading by degree. LP(T, V ) also has the standard grading by order (= number of Lie brackets + 1), which satisfies LP =

m>0 Lm and

[Ln, Lm] ⊆ Ln+m.

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The Lie algebra of classical mechanics

Theorem. LP(T, V ) = Z ⊕ L(T, [T, Z]) where Z = {V, [V, [V, T]], . . .} is an infinite set of degree–0 (potential energy) functions. dim Ln

P(T, V ) ∼ 1

nαn, where α = 1.82542377420108 . . . is the entropy of classical mechanics. (Recall that in the free case, dim Ln(A, B) ∼ 1

n2n (Witt 1936).)

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GF for CM

−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.4 0.7 1 1.15 1.35 1.55 1.75 2 2.3 0.7 1

• ex(t)

where x(t) = ∞

n=0 dim Ln P(T, V )tn.

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Classification of dynamical systems

Sets of dynamical systems can form a semigroup (examples: systems with Lyapunov functions, systems which contract volume) symmetric space closed under (ϕ, ψ) → ϕψ−1ϕ (example: maps with a reversing symmetry ϕ−1 = RϕR−1) group (examples: all diffeomorphisms, symplectic maps, volume-preserving maps) with linearizations to a Lie wedge, Lie triple, and Lie algebra respectively.

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The big picture

• 1. to find all groups, semigroups, and symmetric spaces of

diffeomorphisms

• 2. to find their normal forms
• 3. to find a way to detect the structure in a given system
• 4. to study their relationships under intersection
• 5. to study the dynamics of their perturbations
• 6. to determine the characteristic dynamics and invariants of

each class

• 7. to develop good (i.e. simple, fast, stable) integrators for

each class

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What was known in 1895

conformal all differential equations all "primitive" equations symplectic volume preserving "nonprimitive" equations all contact conformal volume preserving symplectic

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What is known in 2003

Lie was right!

all differential equations all "primitive" equations symplectic volume preserving "nonprimitive" equations all contact conformal volume preserving symplectic conformal

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Nonprimitive groups of diffeomorphisms

preserve some foliation of phase space (leaves map to leaves) may preserve some (e.g. symplectic) structure on the leaves may preserve some structure on the space of leaves may have some complicated interaction between the leaves and the leaf space

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 x y −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 x y −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 x y

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Example

Let G be a Lie group acting on the Poisson manifold (P, {, }). Let H : P → be a G-invariant Hamiltonian. Its flow preserves 4 foliations:

• 1. the level sets of H
• 2. the level sets of the momentum map of G
• 3. the symplectic leaves of P
• 4. the orbits of G

More generally, a diffeomorphism ϕ on a manifold M can preserve any set of foliations, which must contain ∅, M, and be closed under intersections and joins. That is, the set of foliations forms a lattice (generalizes Kodaira & Spencer 1961).

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Lattices with ≤ 3 foliations

not foliate TM D TM D2 D1 not modular not distributive

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The big picture

• 1. to find all groups, semigroups, and symmetric spaces of

diffeomorphisms

• 2. to find their normal forms
• 3. to find a way to detect the structure in a given system
• 4. to study their relationships under intersection
• 5. to study the dynamics of their perturbations
• 6. to determine the characteristic dynamics and invariants of

each class

• 7. to develop good (i.e. simple, fast, stable) integrators for

each class

Thank you for your attention!

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