Geometric Integration of Differential Equations

1. Introduction and ODEs

Chris Budd

Want to simulate a physical system governed

by differential equations

Expect the numerical approximation to have the

same qualitative features as the underlying solution

Traditional approach

Carefully approximate the differential operators in the system Solve the resulting difference equations Monitor and control the local error

Basis of most black box codes and gives excellent results

over moderate computing times

BUT This is a local process and does not pay attention to

the qualitative (global) features of the solution

Geometric Integration

Aims to reproduce qualitative and global features

Some global features

Some qualitative properties:

Conservation laws

Global quantities: Energy, momentum, angular momentum

Flow invariants: Potential vorticity, Casimir functions

Phase space geometry

Symplectic structure

Symmetries

Galilean

Reversal

Scaling: Nonlinear Schrodinger

Lie Group: SO3 (Rigid body)

Asymptotic behaviour

Orderings

Often linked: Noether’s theorem for Lagrangian flows

Conserved quantities:

Symmetries: Rotation, Reflexion, Time reversal, Scaling

Kepler's Third Law

Hamiltonian Angular Momentum

Example: The Kepler Problem

2/3222

2

2/3222

2

)(,

)( yx

y

dt

yd

yx

x

dt

xd

yuxvLyx

vuH

,)(

)(2

12/122

22

),(),(),,(),(, 3/13/2 vuvuyxyxtt

Geometric Integration Aims to preserve a subset of these features Take advantage of powerful global error estimates

(shadowing) Powerful methods for important physical problems

Examples of GI methods

Symplectic and multi-symplectic

Splitting

Lie Group/Magnus

Discrete Lagrangian

Scale Invariant

Examples of GI applications

Molecular and celestial mechanics

Rigid body mechanics

Weather forecasting

Integrable systems (optics)

Self-similar PDEs

Highly oscillatory problems

Example of the traditional and the GI approach:

Integrating the Harmonic Oscillator

Qualitative features: Bounded periodic solutions, time reversal symmetry,

Conserved

Forward Euler method (non GI)

xdt

dyy

dt

dx ,

)(2

1 22 yxH

nnnnnn hXYYhYXX 11 ,

nnnnn HhHYXH )1()(2

1 21

22

Problem: Energy increases, lack of periodicity, lack of symmetry

Backward Euler Method (non GI)

Problem: Energy decreases, lack of periodicity, lack of symmetry

Mid-point rule (a GI method)

1111 , nnnnnn hXYYhYXX

)1/( 21 hHH nn

)(2

),(2 1111 nnnnnnnn XX

hYYYY

hXX

nn HH 1

FE BE

Mid-Point rule

Mid point rule conserves:

Energy

Symmetry

Backward (Modified) Equation Analysis

Solutions are: Bounded, periodic

Phase error proportional to

Discrete equation has an exact solution

Discrete solution shadows the continuous one

)(12

1),sin(),cos( 42

hOh

tYtX nnnn

th2

Symplectic Methods

The mid-point rule behaves well because it conserves the

symplectic structure of the system

Classical Hamiltonian ordinary differential equation:

p

H

dt

dq

q

H

dt

dp

,

0

0,1

I

IJHJ

dt

du),( qpu

Differential equation induces a FLOW )(ut

FLOW MAP is symplectic

JJT ''

Symplecticity places a strong constraint on the flows

1. Preservation of phase space volume (and wedge product)

2. Recurrence

3. No evolution on a low dimensional attractor

4. KAM behaviour for near integrable systems

5. Composition of two symplectic flows is a symplectic flow

Numerical method applied with a constant step size h gives a

map

)( phh hO

h

Traditional analysis:

Show that

GI approach:

Show that is a symplectic map (symplectic method)h

Advantage: Symplectic methods have good ergodic properties

Strong error estimates via backward error analysis

method is exact solution of a perturbed Hamiltonian problem

Symplectic Methods include: Runge-Kutta, Splitting, Variational

Runge-Kutta methods for du/dt = f(u)

There is a large class of implicit symplectic Runge-Kutta methods

c A

b

Construct matrix M jijijijiij bbababM

Method is symplectic if M = 0

Butcher Tableaux

All linear and quadratic invariants conserved LuuT

)2/)(( 11 nnnn uuhfuu

Example: the implicit mid-point rule

All Gauss-Legendre Runge-Kutta methods and associated

collocation methods are symplectic

Symplectic, implicit, symmetric, unconditionally stable,

Conserves linear and quadratic invariants

Splitting and composition methods

Runge-Kutta methods are implicit, but for certain problems we

can construct explicit symplectic methods via splitting

)()( 21 ufufdt

du

h,1h,2Construct flow maps and for and 1f 2f

hhhT ,2,1, 2/,1.22/,1, hhhhS

Compose the split maps

Strang splittingLie-Trotter splitting

Some important results

If and are symplectic, so are

The Campbell-Baker-Hausdorff theorem implies that

h,1 h,2 hS ,hT ,

)( 2, hOhTh )( 3

, hOhSh

If H(u) = T(p) + V(q)

The splittings lead directly to two important numerical methods

)(),( 111 nnnnnn pThqqqVhpp

)(2

),(),(2 12/112/112/1 nnnnnnnnn qV

hpppThqqqV

hpp

Symplectic Euler SE

Stormer-Verlet SV (Leapfrog)

Symplectic, explicit, non-symmetric, order 1

Symplectic, explicit, symmetric, order 2

Unstable for large step size

There are higher order, explicit, splitting methods due to

Yoshida, Blanes.

Apply to the Kepler problem

SV

SE

FE

Global error

H error

FE

SE

Method Global error H error L error

FE t^2 h t h t h

SE t h h 0

SV t h^2 h^2 0

NOTE: Kepler’s third law is NOT conserved by these methods …

see the next talk!

Backward Error Analysis

Up to an exponentially small (In h) error

the solutions of a symplectic method of order p are the

discrete samples of a solution of a related Hamiltonian

differential equation with Hamiltonian

...)()()()( 11

uHhuHhuHuH pp

pp

h

Can construct the perturbed Hamiltonian explicitly

H error remains bounded for all times

Doesn’t apply if h varies!

)( /* hheOE

Example: A problem in structural mechanics

22

12

qp

H

)(12

12

)( 2

2

22

21 hO

q

pqhq

phOhHHH h

Discrete Euler Beam

Small h limit

Hamiltonian for Symplectic Euler discretisation = original problem

hH

1hHH

h = 0.05 h = 1.1 h = 2.2

Symmetry Group Methods

Important class of GI methods are used to solve problems

with Lie Group Symmetries (deep conservation laws)

gAGuuutAdt

du ,,),(

G: (matrix) Lie group g: Lie algebra

Eg. G = SO3 (rotations), g = so3 (skew symmetry)

Spo

tty

dog

Can a numerical method ensure that the solution remains in G?

Rigid body mechanics, weather forecasting, quantum mechanics, Lyapunov

exponents, QR factorisation

Idea: Do all computations in the Lie Algebra (linear space)

And map between this and the Lie Group (nonlinear space)

G

g

nU 1nU

n1n

numerical method

)2/()2/()(,!

)exp( 1 AIAIAcayn

AA

n

Examples of maps from g to G

General g , G g = so3, G = SO3

satisfies the dexpinv equation

),(! n

ll UetAadl

B

dt

d

Integrate the dexpinv equation numerically

Conserve the group structure by making sure that all

numerical approximations to the dexpinv equation always

lie in the Lie algebra

Fine provided method uses linear operations and commutators

Runge-Kutta/Munthe-Kaas (RKMK) methods use this approach

tt

ddAAdAt0 0 21120

1

)](),([2

1)()(

...)]()],(),([[4

1320 0 0 1123

1 2

dddAAAt

utAdt

du)(

Magnus series methods:

Magnus series:

Obtain method by series truncation and careful calculation of the

commutators

VERY effective for Highly Oscillatory Problems [Iserles]

3

02/2/)cos(

2/0

2/)cos(0

)( so

tt

tt

tt

tA

Eg. Evolution on the surface of the sphere

uuI T invariant

FE RK

RKMK Magnus