Transcript Some Geometric integration methods for PDEs Chris Budd (Bath) Have a PDE with solution u(x,y,t) ut  F (u, ux , u y.

Some Geometric integration methods for
PDEs
Chris Budd (Bath)
Have a PDE with solution u(x,y,t)
ut  F (u, ux , u y , uxx , u yy , ...)
Variational structure
Symmetries linking space and time
Conservation laws
Maximum principles
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
Variational Calculus

u    G
 
,
t  x  u
 0
dG
 0,
dt
G   G dx
dG
 1
0
dt
Hamiltonian system
u
 SH ,
t
H   H dx ,
dH
0
dt
u
 u  u 3
t
u G

,
t u
G   G dx   
u
2
2
u4
 dx
4
u
2
i
 u  u u  0
t
u
H
 i
,
t
u
H   H dx   u 
2
u
4
2
dx
dG
  ut2 dx  0,
dt
u    as t  t
NLS is integrable in one-dimension,
In higher dimensions
dH
 0,
dt
u

  as t  t ,
Can we capture this behaviour?
u 2 C
Discrete Variational Calculus [B,Furihata,Ide]

u    G
 
,
t  x  u
G   G(u, x) dx
Gd  T Gd (U )k x
Gd (Ukn )  G(u(nt, kx))
Gd (U )  Gd (V )  T 
Gd
 (U ,V ) k
(U k  Vk )x
Gd   f l (U k ) g ( U k ) g ( U k ),

l

k

l

k
Gd
U k 1  U k
 (U k ) 
x

k
dfl
g l ( kU k ) g l ( kU k )  g l ( kVk ) g l ( kVk )

 (U ,V ) k
2
l d (U k , Vk )
  kWl  (U ,V ) k   kWl  (U , V ) k

u    G
 
,
t  x  u
G   G(u, x) dx
U kn1  U kn
Gd
( )
 k
t
 (U n1 ,U n ) k
Gd (U n 1 )  Gd (U n )  T 
0
Gd
n 1
n
( ) 

(
U

U
)

x

T

   (U n1 ,U n )
 (U n 1 ,U n ) k
k

  0,
Gd
0
 1

Gd

xt
n 1
n

(
U
,
U
)
k

Example:
u
 u xx  u 3 ,
t
u x (0)  u x (1)  0,

u x2 u 4
G 
2
4
   
U kn1  U kn 1 ( 2) n1
1 n1 3
n
n 1 2
n
n 1
n 2
  Uk Uk  Uk
 U k U k  U k U k  U kn
t
2
4


 

Implementation :
• Predict solution at next time step using a standard
implicit-explicit method
• Correct using a Powell Hybrid solver
3
Problem: Need to adaptively update the time step
u
 u xx  u 3 ,
t
u   t  t
Balance the scales
1
u U u  T  2
U
t  T t,

   
U kn 1  U kn
1 ( 2 ) n 1
1
n
n 1 3
n 1 2
n
n 1
n 2
   Uk Uk  Uk
 U k U k  U k U k  U kn
t n
2
4


 
 
 
 max U n 2 
 t n
t n 1  
2
 max U n 1 


t   t n

3
t
n
G
U
G
U
n
t
u
x
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
Extend the idea of balancing the scales in d dimensions
u
 u xx  u yy  u 3 ,
t
u  , t  t 
t  T t,
u  U u , ( x, y )  L ( x, y )
1
1
 T  2 , L  , T  (t  t )
U
U
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  C into a physical domain  P
F
C ( , )
 P ( x, y)
Use a strategy for computing the mesh mapping function
F which is simple, fast and takes geometric properties
into account [cf. Image registration]
Introduce a mesh potential
Q( , , t )
( x(t ), y(t ),..)   Q  (Q , Q ,..)
Geometric scaling
 Q
 ( x, y )
H (Q) 
 det
( , )
 Q
Q 

Q 

Q

2
Q Q  Q
1D
Control scaling via a measure
2D
M (u, ux , u y ,..)
Evolve mesh by solving a MK based PDE
 I   Qt  M (Q) H (Q) 
1/ d
Spatial smoothing
(Invert operator
using a spectral
method)
Averaged
measure
(PMA)
Ensures right-handside scales like P in
dD to give global
existence
Parabolic Monge-Ampere equation PMA
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
t  Tt, ( x, y)  L( x, y), u  Uu
( x, y)  L( x, y)  Q  LQ
L
Qt  Qt
T
H ( LQ)1/ d  LH (Q)1/ d
PMA is scale invariant provided that
M ( LQ)1/ d  M (Uu( L( x, y),Tt )1/ d  T 1M (u( x, y, t ))1/ d
Extremely useful property when working with
PDEs which have natural scaling laws
Example: Parabolic blow-up in d-D
ut  uxx  u yy  u
Scale:
3
u   t  t ,
T  (t  t ), U  T
1 / 2
( x, y)  ( x* , y* )
, L T
1/ 2
log(T )
M (T 1/ 2u)1/ d  T 1M (u)1/ d  M (u)  u 2d
Regularise:
M ( X , Y , t )  u( X , Y , t )d   u( X , Y , t )d d d X
1/ 2
Solve in PMA parallel with the PDE
10
10^5
Solution:
Y
Mesh:
ut  uxx  u yy  u3
X
Solution in the computational domain
10^5


NLS in 1-D