Adaptivity and symmetry for ODEs and PDEs

Chris Budd

Talk will look at

• variable step size adaptive methods for

ODES• scale invariant adaptive methods for PDES

Basic Philosophy …..

• ODES and PDEs develop structures

on many time and length scales

• Structures may be uncoupled (eg. Gravity waves

and slow weather evolution) and need multi-scale

methods

• Or they may be coupled, typically through (scaling)

symmetries and can be resolved using adaptive

methods

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

Stormer

Verlet

Symplectic

Euler

Forward

Euler

Global error

H error

FE

SV

t

Main error

Larger error at close approaches

Kepler’s third law is not respected

Adaptive time steps are highly desirable for

accuracy and symmetry

But …

Adaptivity can destroy the symplectic shadowing

structure [Calvo+Sanz-Serna]

Adaptive methods may not be efficient as a splitting

method

AIM: To construct efficient, adaptive, symplectic

methods EASY which respect symmetries

H error

t

The Sundman transform introduces a continuous

adaptive time step.

IDEA: Introduce a fictive computational time

),( qpgd

dt

qp Hdt

dpH

dt

dq ,

Hamiltonian ODE system:

),( qpg SMALL if solution requires small time-steps

qp Hqpgd

dpHqpg

d

dq),(,),(

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

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

),( qpgd

dt

Can make Hamiltonian via the Poincare Transform

New variables

Hamiltonian

Rescaled system for p,q and t

Now solve using a Symplectric ODE solver

Choice of the scaling function g(q)

Performance of the method is highly dependent on

the choice of the scaling function g.

Approach: insist that the performance of the

numerical method when using the computational

variable should be independent of the scale of the

solution and that the method should respect the

symmetries of the ODE

),...,( 1 Nii uuf

dt

du

0,, uutt i

)()( tutu iii

The differential equation system

Is invariant under scaling if it is unchanged by the symmetry

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: 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 ,, If symmetry

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

0

Numerical scheme is scale-invariant if

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 change

H Error

nopt 2

1

2

3

Method order

Surprisingly

sharp!!!

Scale invariant methods for PDES

These methods extend naturally to PDES with

scaling and other symmetries

uuyxyxtt

uuufu xxt

),,(),(,

,...),,,(

.),(,0),0(,)(

,)(

,),.(

,0

,

,

2

3

3

vttsutuvquu

uuu

vuvvvuuu

uuuiu

uuu

uuu

xm

t

xxm

t

tt

t

xxxxt

xxt

Examples

Parabolic blow-up

High-order blow-up

NLS

Chemotaxis

PME

Rainfall

Need to continuously adapt in time and space

Introduce spatial analogue of the fictive time

Adapt spatially by mapping a uniform mesh from a computational domain into a physical domain

Use a strategy for computing the mesh which takes symmetries into account

C P

),( C ),( yxP

Introduce a mesh potential ),,( tQ

,..),(),..)(),(( QQQtytx

DQQQ

DQ

QQ

QQyxQH

2

1

det),(

),()(

2

,..),,( yx uuuM

Geometric scaling

Control scaling via a measure

dt QHQMQI /1)()(

Spatial smoothing

(Invert operator using a spectral method)

Averaged measure

Ensures right-hand-side scales like P in d-dimensions to give global existence

Parabolic Monge-Ampere equation PMA

(PMA)

Evolve mesh by solving a MK based PDE

Because PMA is based on a geometric approach, it has natural symmetries

1. System is invariant under translations and rotations

2. For appropriate choices of M the system is invariant under scaling symmetries

LQQyxLyx ),(),(

ddtt QHQLMLLQHQLMQ

T

LQ /1/1 ))()(())()((,

PMA is scale invariant provided that

ddd tyxuMTTtyxLUuMQLM /11/1/1 )),,(())),,((()(

UuuyxLyxTtt ),,(),(,

2/12/12/1 )log(,),( TTLTUttT

),(),(, ** yxyxttu

3uuuu yyxxt

XdtYXutYXutYXM ddd 22 ),,(),,(),,(

Example: Parabolic blow-up in d dimensions

ddd uuMuMTuTM 2/11/12/1 )()()(

Scale:

Regularise:

uuyxyxtttt 2/12/1 ),,(),(),()(

Basic approach

• Discretise PDE and PMA in the computational domain

• Solve the coupled mesh and PDE system either

(i) As one large system (stiff!) or

(ii) By alternating between PDE and mesh

Method admits exact discrete self-similar solutions

solve PMA simultaneously 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 PDEsNow extending it to CFD problems: Eady, Bousinessq