Geometric Integration of Differential Equations

2. Adaptivity, scaling and PDEs

Chris Budd

Previous lecture considered constant step size

Symplectic methods for Hamiltonian ODEs

Now we will look at variable step size adaptive

methods for ODES

We will extend them to scale invariant methods for

a wide class of PDES

Then look at more general symplectic methods for

Hamiltonian PDEs

Conserved quantities:

Symmetries: Rotation, Reflexion, Time reversal, Scaling

Kepler's Third Law

Hamiltonian Angular Momentum

The need for adaptivity: 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

Kepler orbits

SV

SE

FE

Global error

H error

FE

SV

t

Main error

Larger error at close approaches

Kepler’s third law is not respected

Problems with fixed step Symplectic methods

Advantages of fixed step Symplectic methods

Conservation of L, near conservation of H, shadowing,

efficient high order splitting methods.

Adaptive time steps are highly desirable

But …

Adaptivity can destroy the shadowing structure [Sanz-Serna]

Adaptive methods may not be efficient as a splitting method

AIM: To construct efficient, adaptive, symplectic methods

EASY

H error

t

Symplectic methods and the Sundman transform

The Sundman transform is a means of introducing a

continuous adaptive time step.

IDEA: Introduce a fictive computational time

),( qpgd

dt

qp Hdt

dpH

dt

dq ,

Hamiltonian system:

),( qpg

))0(),0((, 0 qpHHqtp tt

),(),( tptqq qHggpd

dqqHgVg

d

dp

SMALL if solution requires small time-steps

BUT .. rescaling of system is NOT initially Hamiltonian.

Use Poincare transformation for

0, d

dqg

d

dp tt

)),()(,(),,,( ttt qqpHqpgqqppK

Hamiltonian

)(2

1qVppH T

Good news: Rescaled system is Hamiltonian.

Bad news: Hamiltonian is not separable

Can’t use efficient splitting methods

Method one [Hairer] Use an implicit Symplectic method

Method two [Reich, Leimkuhler,Huang] Use an efficient

symmetric (non-symplectic ) adaptive Verlet method

Method three [B, Blanes] Use a canonical transform to

obtain a separable Hamiltonian

)(~

)(~

qVpTK

Canonical transformation: Introduce new variables (P,Q) for

which we have a separable Hamiltonian system.

))()(()( tT qqVqgppqgK

PpQQq T )('),(

2)'()( g

Consider the special scalar case:

Theorem: The following transformation is canonical

Now find by solving

Choice of the scaling function g(q)

Performance of the method is highly dependent on the

choice of the scaling function g.

There are many ways to do this!

One approach is to insist that the performance of the

numerical method when using the computational variable

should be independent of the scale of the solution

),...,( 1 Nii uuf

dt

du

0,, uutt i

)()( tutu iii

The differential equation system

Is invariant under scaling if it is unchanged by the transformation

It generically admits particular self-similar solutions satisfying

eg. Kepler’s third law relating planetary orbits

),...,(),...,( 111

NN uuguug N

Theorem [B, Leimkuhler,Piggott] If the scaling function

satisfies the functional equation

Then

Two different solutions of the original ODE mapped onto

each other by the scaling transformation are the same

solution of the rescaled system scale invariant

A discretisation of the rescaled system admits a discrete

self-similar solution which uniformly approximates the true

self-similar solution for all time

Example 1: Kepler problem in radial coordinates

A planet moving with angular momentum with radial

coordinate r = q and with dr/dt = p satisfies a Hamiltonian

ODE with Hamiltonian

2

2 1

2 qq

pH

ppqqtt 3/13/2 ,,

3/2)( tTCq

If this is invariant under

2/33/2 )()()( qqgqgqg

0

Self-similar collapse solution

If there are periodic solutions with close approaches

Hard to integrate with a non-adaptive scheme

10

q

t

2

2/3

1

0

)(

qqg

Consider calculating them using the scaling

No scaling

Levi-Civita scaling

Scale-invariant

Constant angle

H Error

H Error

nopt 2

1

2

3

Method order

t

P

Q

1 1.5 1.8

Example 2: Motion of a satellite around an oblate planet

Integrable if

Levi-Civita scaling works best in this case

If then scale invariant scaling is best for

eccentric orbits

2

21

3

22

21 1

2

1)(

2

1

r

q

rrppH

0

10

L-C SI

eccentricity

Extension of scale invariant methods to PDES

These methods extend naturally to PDES with a

scaling invariance

uuxxtt

uuufu xxt

,,

,...),,,(

Idea: Introduce a computational coordinate

And a differential equation linking to X

dd

t

d

PDuMX

uMP

PX

/12

/1

)()(

)(

Mesh potential P

Monitor function M (large where mesh points need to cluster)

Parabolic Monge-Ampere equation PMA

Choose the monitor function M(u) by insisting that the system

should be invariant under changes of spatial and temporal scale

212/1

2/12/1

3

1

)()()(

,,

)()(

uuMuMuM

uuxxtt

uuu

uMuM

xt

Example: Parabolic blow-up equation

Scaling:

Monitor:

Ttasu

Solve PMA in parallel with the PDE3uuu Xt

Mesh:

Solution:

XY

10 10^5

Solution in the computational domain

10^5

12

Same approach works well for the Chemotaxis eqns, Nonlinear Schrodinger eqn, Higher order PDEs

More general geometric integration methods for PDES

Geometric integration methods for PDES are much less well developed than for ODEs as PDES have a very rich structure and many conservation laws and it is hard to preserve all of this under discretisation

Hamiltonian: NLS, KdV, Euler eqns

Lagrangian structure:

Scaling laws: NLS, parabolic blow-up, Porous medium eqn

Conservation laws and integrability: NLS

Have to choose what to preserve under discretisation:

Variational integrators, scale-invariant, multi-symplectic

Example: Multi-symplectic methods for Hamiltonian PDEs

[Bridges, Reich, Moore,Frank, Marsden, Patrick, Schkoller,McLachlan,Ascher]

SzLzK

Szz

uFPwwv

Spwvuz

wpwuvu

ufuu

zxt

zxt

xt

xxtt

0001

0000

0000

1000

0000

0000

0001

0010

)(22

),,,,(

,,,

)(

22

NLWE

Many PDEs have this multi-symplectic form

Shallow water, NLS, KdV, Boussinesq

They typically have local conservation laws of the form

0 xt FE

IDEA Discretise these equations using a symplectic method in t and a symplectic method in x

Eg. Use the implicit mid-point rule

jiji

ji

ji

z

ji

ji

ji

ji

ZZZ

ZSx

ZZL

t

ZZK

12/1

2/12/1

2/11

2/12/12/1

1

2

1

)(

Preissman/Keller Box Scheme

Preissman/Keller Box Scheme

• Preserves conservation laws arising from linear symmetries

• Preserves energy and momentum for linear PDES

• Gives correct dispersion relation for linear equations

• Not much known for nonlinear problems

Study using backward error analysis: modified equation has a multi-symplectic structure, but don’t get exponentially small estimates.

Conclusion

Geometric integration has proved to be a powerful tool for integrating ODEs with many different scales

Its potential for PDES is still being developed, but it could have a significant impact on problems such as weather forecasting

It is an area where pure mathematicians, applied mathematicians, numerical analysts, physicists etc must all work together