Geometric Integration of Differential Equations

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Geometric Integration of Differential
Equations 2. Adaptivity, scaling and PDEs
Chris Budd
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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
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The need for adaptivity the Kepler problem
Conserved quantities
Hamiltonian
Angular Momentum
Symmetries Rotation, Reflexion, Time reversal,
Scaling
Kepler's Third Law
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Kepler orbits
FE
SE
SV
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FE
Global error
SV
H error
Main error
t
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Advantages of fixed step Symplectic methods
Conservation of L, near conservation of H,
shadowing, efficient high order splitting methods.
Problems with fixed step Symplectic methods
Larger error at close approaches
Keplers third law is not respected
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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
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H error
t
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Symplectic methods and the Sundman transform
Hamiltonian system
The Sundman transform is a means of introducing a
continuous adaptive time step.
IDEA Introduce a fictive computational time
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SMALL if solution requires small time-steps
BUT .. rescaling of system is NOT initially
Hamiltonian.
Use Poincare transformation for
Hamiltonian
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Good news Rescaled system is Hamiltonian.
Bad news Hamiltonian is not separable Cant 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
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Canonical transformation Introduce new variables
(P,Q) for which we have a separable Hamiltonian
system.
Consider the special scalar case
Theorem The following transformation is canonical
Now find by solving
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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
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The differential equation system
Is invariant under scaling if it is unchanged by
the transformation
eg. Keplers third law relating planetary orbits
It generically admits particular self-similar
solutions satisfying
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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
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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
If this is invariant under
Self-similar collapse solution
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If there are periodic
solutions with close approaches Hard to
integrate with a non-adaptive scheme
q
t
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Consider calculating them using the scaling
No scaling Levi-Civita scaling Scale-invariant Con
stant angle
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H Error
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H Error
Method order
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1 1.5
1.8
Q
P
t
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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
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L-C
SI
eccentricity
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Extension of scale invariant methods to PDES
These methods extend naturally to PDES with a
scaling invariance
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Idea Introduce a computational coordinate And a
differential equation linking to X
Mesh potential P
Monitor function M (large where mesh points need
to cluster)
Parabolic Monge-Ampere equation PMA
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Choose the monitor function M(u) by insisting
that the system should be invariant under changes
of spatial and temporal scale
Example Parabolic blow-up equation
Scaling Monitor
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Solve PMA in parallel with the PDE
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105
Solution
Y
X
Mesh
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Solution in the computational domain
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Same approach works well for the Chemotaxis eqns,
Nonlinear Schrodinger eqn, Higher order PDEs
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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
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Example Multi-symplectic methods for
Hamiltonian PDEs Bridges, Reich, Moore,Frank,
Marsden, Patrick, Schkoller,McLachlan,Ascher

NLWE
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Many PDEs have this multi-symplectic form
Shallow water, NLS, KdV, Boussinesq
They typically have local conservation laws of
the form
IDEA Discretise these equations using a
symplectic method in t and a symplectic method in
x
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Eg. Use the implicit mid-point rule
Preissman/Keller Box Scheme
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• 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
dont get exponentially small estimates.
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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