Some Geometric integration methods for PDEs

Chris Budd (Bath)

Have a PDE with solution u(x,y,t)

Variational structure

Symmetries linking space and time

Conservation laws

Maximum principles

)...,,,,,( yyxxyxt uuuuuFu

Cannot usually preserve all of the structure and

Have to make choices

Not always clear what the choices should be

BUT

GI methods can exploit underlying mathematical links between different structures

dxGu

G

xt

uG,

01,00 dt

d

dt

d GG

Variational Calculus

0,,

dt

ddxHSH

t

u HH

Hamiltonian system

dxuu

dxGu

G

t

u

42,

42

G

3uut

u

02

uuut

ui

dxu

udxHu

Hi

t

u

2,

42

H

ttudxu

dt

dt as,02G

Cuttasudt

d 2

,,0H

NLS is integrable in one-dimension,

In higher dimensions

Can we capture this behaviour?

)),(()( xktnuGUG nkd xUGT kdd )(G

xVUVU

GTVU kk

k

ddd )(

),()()(

GG

Discrete Variational Calculus [B,Furihata,Ide]

dxxuGu

G

xt

u),(, G

x

UUUUgUgUfG kkkkkklkklkld

1)(),()()(

klkklk

kklkklkklkkl

l kk

l

k

d

VUWVUW

VgVgUgUg

VUd

df

VU

G

),(),(

2

)()()()(

),(),(

knn

dk

nk

nk

UU

G

t

UU

),( 1)(

1

10,00

),(),()(

),()()(

11)(1

11

tx

UU

G

UU

GTxUU

UU

GTUU

knn

d

knn

dnn

knn

dnd

nd GG

dxxuGu

G

xt

u),(, G

Example:

42,0)1()0(,

423 uu

Guuuut

u xxxxx

32121311)2(1

4

1

2

1 nk

nk

nk

nk

nk

nk

nk

nk

nk

nk UUUUUUUUt

UU

Implementation :

• Predict solution at next time step using a standard implicit-explicit method

• Correct using a Powell Hybrid solver

n

nn

n

n

nk

nk

nk

nk

nk

nk

nk

nk

n

nk

nk

tt

tU

Ut

UUUUUUUUt

UU

21

2

1

32121311)2(1

max

max

4

1

2

1

ttuuut

uxx ,3

Problem: Need to adaptively update the time step

Balance the scales

2

1,

UTuUutTt

t

n

G

U

G

U

n

t

x

u

Some issues with using this approach for singular problems

• Doesn’t naturally generalise to higher dimensions

• Doesn’t exploit scalings and natural (small) length scales

• Conservation is not always vital in singular problems

Peak may not contribute asymptotically NLS

ttuuuut

uyyxx ,,3

)(,1

,1

),(),(,,

2ttT

UL

UT

yxLyxuUutTt

Extend the idea of balancing the scales in d dimensions

Need to adapt the spatial variable

Use r-refinement to update the spatial mesh

Generate a mesh by mapping a uniform mesh from a computational domain into a physical domain

Use a strategy for computing the mesh mapping function F which is simple, fast and takes geometric properties into account [cf. Image registration]

F

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

d

t QHQMQI/1

)()(

Spatial smoothing

(Invert operator using a spectral method)

Averaged measure

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

Parabolic Monge-Ampere equation PMA

(PMA)

Evolve mesh by solving a MK based PDE

Geometry of the method

Because PMA is based on a geometric approach, it performs well under certain geometric transformations

1. System is invariant under translations and rotations

2. For appropriate choices of M the system is invariant under natural scaling transformations of the form

UuuyxLyxTtt ),,(),(,

LQQyxLyx ),(),(

ddtt QLHLQHQ

T

LQ /1/1 )()(

PMA is scale invariant provided that

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

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

),(),(, ** yxyxttu 3uuuu yyxxt

Extremely useful property when working with PDEs which have natural scaling laws

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

Example: Parabolic blow-up in d-D

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

Scale:

Regularise:

Solve in PMA parallel with the PDE

3uuuu yyxxt

Mesh:

Solution:

XY

10 10^5

Solution in the computational domain

10^5

NLS in 1-D