View
33
Download
4
Category
Preview:
DESCRIPTION
Geometric Integration of Differential Equations 2. Adaptivity, scaling and PDEs Chris Budd. Previous lecture considered constant step size Symplectic methods for Hamiltonian ODEs. Now we will look at variable step size adaptive methods for ODES - PowerPoint PPT Presentation
Citation preview
Geometric Integration of Differential Equations
2. Adaptivity, scaling and PDEs
Chris Budd
Previous lecture considered constant step size
Symplectic methods for Hamiltonian ODEs
Now we will look at variable step size adaptive
methods for ODES
We will extend them to scale invariant methods for
a wide class of PDES
Then look at more general symplectic methods for
Hamiltonian PDEs
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
SV
SE
FE
Global error
H error
FE
SV
t
Main error
Larger error at close approaches
Kepler’s third law is not respected
Problems with fixed step Symplectic methods
Advantages of fixed step Symplectic methods
Conservation of L, near conservation of H, shadowing,
efficient high order splitting methods.
Adaptive time steps are highly desirable
But …
Adaptivity can destroy the shadowing structure [Sanz-Serna]
Adaptive methods may not be efficient as a splitting method
AIM: To construct efficient, adaptive, symplectic methods
EASY
H error
t
Symplectic methods and the Sundman transform
The Sundman transform is a means of introducing a
continuous adaptive time step.
IDEA: Introduce a fictive computational time
),( qpgd
dt
qp Hdt
dpH
dt
dq ,
Hamiltonian system:
),( qpg
))0(),0((, 0 qpHHqtp tt
),(),( tptqq qHggpd
dqqHgVg
d
dp
SMALL if solution requires small time-steps
BUT .. rescaling of system is NOT initially Hamiltonian.
Use Poincare transformation for
0, d
dqg
d
dp tt
)),()(,(),,,( ttt qqpHqpgqqppK
Hamiltonian
)(2
1qVppH T
Good news: Rescaled system is Hamiltonian.
Bad news: Hamiltonian is not separable
Can’t use efficient splitting methods
Method one [Hairer] Use an implicit Symplectic method
Method two [Reich, Leimkuhler,Huang] Use an efficient
symmetric (non-symplectic ) adaptive Verlet method
Method three [B, Blanes] Use a canonical transform to
obtain a separable Hamiltonian
)(~
)(~
qVpTK
Canonical transformation: Introduce new variables (P,Q) for
which we have a separable Hamiltonian system.
))()(()( tT qqVqgppqgK
PpQQq T )('),(
2)'()( g
Consider the special scalar case:
Theorem: The following transformation is canonical
Now find by solving
Choice of the scaling function g(q)
Performance of the method is highly dependent on the
choice of the scaling function g.
There are many ways to do this!
One approach is to insist that the performance of the
numerical method when using the computational variable
should be independent of the scale of the solution
),...,( 1 Nii uuf
dt
du
0,, uutt i
)()( tutu iii
The differential equation system
Is invariant under scaling if it is unchanged by the transformation
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 1: 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 ,,
3/2)( tTCq
If this is invariant under
2/33/2 )()()( qqgqgqg
0
Self-similar collapse solution
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
H Error
H Error
nopt 2
1
2
3
Method order
t
P
Q
1 1.5 1.8
Example 2: Motion of a satellite around an oblate planet
Integrable if
Levi-Civita scaling works best in this case
If then scale invariant scaling is best for
eccentric orbits
2
21
3
22
21 1
2
1)(
2
1
r
q
rrppH
0
10
L-C SI
eccentricity
Extension of scale invariant methods to PDES
These methods extend naturally to PDES with a
scaling invariance
uuxxtt
uuufu xxt
,,
,...),,,(
Idea: Introduce a computational coordinate
And a differential equation linking to X
dd
t
d
PDuMX
uMP
PX
/12
/1
)()(
)(
Mesh potential P
Monitor function M (large where mesh points need to cluster)
Parabolic Monge-Ampere equation PMA
Choose the monitor function M(u) by insisting that the system
should be invariant under changes of spatial and temporal scale
212/1
2/12/1
3
1
)()()(
,,
)()(
uuMuMuM
uuxxtt
uuu
uMuM
xt
Example: Parabolic blow-up equation
Scaling:
Monitor:
Ttasu
Solve PMA in parallel 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 PDEs
More general geometric integration methods for PDES
Geometric integration methods for PDES are much less well developed than for ODEs as PDES have a very rich structure and many conservation laws and it is hard to preserve all of this under discretisation
Hamiltonian: NLS, KdV, Euler eqns
Lagrangian structure:
Scaling laws: NLS, parabolic blow-up, Porous medium eqn
Conservation laws and integrability: NLS
Have to choose what to preserve under discretisation:
Variational integrators, scale-invariant, multi-symplectic
Example: Multi-symplectic methods for Hamiltonian PDEs
[Bridges, Reich, Moore,Frank, Marsden, Patrick, Schkoller,McLachlan,Ascher]
SzLzK
Szz
uFPwwv
Spwvuz
wpwuvu
ufuu
zxt
zxt
xt
xxtt
0001
0000
0000
1000
0000
0000
0001
0010
)(22
),,,,(
,,,
)(
22
NLWE
Many PDEs have this multi-symplectic form
Shallow water, NLS, KdV, Boussinesq
They typically have local conservation laws of the form
0 xt FE
IDEA Discretise these equations using a symplectic method in t and a symplectic method in x
Eg. Use the implicit mid-point rule
jiji
ji
ji
z
ji
ji
ji
ji
ZZZ
ZSx
ZZL
t
ZZK
12/1
2/12/1
2/11
2/12/12/1
1
2
1
)(
Preissman/Keller Box Scheme
Preissman/Keller Box Scheme
• Preserves conservation laws arising from linear symmetries
• Preserves energy and momentum for linear PDES
• Gives correct dispersion relation for linear equations
• Not much known for nonlinear problems
Study using backward error analysis: modified equation has a multi-symplectic structure, but don’t get exponentially small estimates.
Conclusion
Geometric integration has proved to be a powerful tool for integrating ODEs with many different scales
Its potential for PDES is still being developed, but it could have a significant impact on problems such as weather forecasting
It is an area where pure mathematicians, applied mathematicians, numerical analysts, physicists etc must all work together
Recommended