Post on 21-Dec-2015
transcript
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
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