# Slide 1

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```					Some Geometric integration methods for
PDEs

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

ut  F (u , u x , u y , u xx , u yy , ... )

Variational structure
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
Variational Calculus


u    G
        ,          G   G dx
t  x  u

dG                   dG
 0         0,          1    0
dt                   dt

Hamiltonian system

u                               dH
 SH ,          H   H dx,       0
t                                dt
u
 u  u 3
t

u
2
u G                                      u4
   ,       G   G dx                dx
t u                            2         4

u
i  u  u u  0
2

t
4
u      H                                     u
 i          H   H dx   u 
2
,                                           dx
t      u                                     2
dG
  ut2 dx  0,    u    as t  t
dt

NLS is integrable in one-dimension,
In higher dimensions

dH
 0,        u   
  as t  t ,   u 2 C
dt

Can we capture this behaviour?
Discrete Variational Calculus [B,Furihata,Ide]

u    G
        ,                     G   G(u, x) dx
t  x  u
Gd (U kn )  G (u (nt , kx))                      Gd  T Gd (U )k x

Gd
Gd (U )  Gd (V )  T                       (U k  Vk )x
 (U , V ) k

U k 1  U k
Gd   f l (U k ) g ( U k ) g ( U k ),
                      
 (U k ) 


x
l     k           l      k                    k

Gd             dfl       gl ( kU k ) gl ( kU k )  gl ( kVk ) g l ( kVk )

 (U ,V ) k  l d (U k , Vk )                              2
  kWl  (U ,V ) k   kWl  (U ,V ) k

u    G
        ,                                G   G(u, x) dx
t  x  u

U kn1  U kn    ( )      Gd
 k
t                (U n1 ,U n ) k

Gd                                              Gd                  Gd
Gd (U n1 )  Gd (U n )  T                      (U n1  U n )x  T   ( ) 
  (U n1 ,U n )     
  (U n 1 ,U n ) xt
 (U n1 ,U n ) k                                                k                    k

0         0,              0        1
Example:

u                                                     2
ux u 4
 u xx  u 3 ,       u x (0)  u x (1)  0,     G 
t                                                    2   4

U kn 1  U kn 1 ( 2) n 1
t       2

  Uk Uk  Uk
n 1
4

n 1 3
      
n 1 2 n

n 1 n 2
 U k U k  U k U k  U kn   
3

Implementation :
• Predict solution at next time step using a standard
implicit-explicit method
• Correct using a Powell Hybrid solver
Problem: Need to adaptively update the time step

u
 u xx  u 3 ,         u   t  t
t

Balance the scales
1
t  T t,        u U u  T  2
U

U kn 1  U kn
t n

1 ( 2 ) n 1
   Uk Uk  Uk
2
n
1
4

n 1 3
 
n 1 2
  n 1
 U k U k  U k U k  U kn
n
   
n 2       3

 
 max U n 2 
t n 1                 t n
 
 max U n 1

2 

t   t n
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 ,     u  , t  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

 ( x, y )       Q   Q 
H (Q)              det
Q

 ( , )            Q 

   Q     1D
   Q Q  Q
2
2D

Control scaling via a measure               M (u , u x , u y ,..)
Evolve mesh by solving a MK based PDE

 I   Qt  M (Q) H (Q)
1/ d

(PMA)

Spatial smoothing
Averaged   Ensures right-hand-
(Invert operator    measure    side scales like P in
using a spectral               dD to give global
method)                        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                        H ( LQ)1/ d  LH (Q)1/ d
T

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     3
u   t  t ,             ( x, y )  ( x * , y * )

1/ 2
T  (t  t ), U  T                   , L T
1/ 2
Scale:
1/ 2
log(T )

1 / 2               1
M (T             1/ d
u)        T M (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
Solve in PMA parallel with the PDE   ut  uxx  u yy  u 3
10                                  10^5
Solution:

Y               X

Mesh:
Solution in the computational domain

10^5

          
NLS in 1-D

```
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