Adaptivity and symmetry for ODEs and PDEs

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Adaptivity and symmetry for ODEs and
PDEs Chris Budd
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• Basic Philosophy ..
• ODES and PDEs develop structures
• on many time and length scales
• Structures may be uncoupled (eg. Gravity waves
  and slow weather evolution) and need multi-scale
  methods
• Or they may be coupled, typically through
  (scaling) symmetries and can be resolved using
  adaptive methods

• Talk will look at
• variable step size adaptive methods for ODES
• scale invariant adaptive methods for 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
Forward Euler
Symplectic Euler
Stormer Verlet
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FE
Global error
SV
Main error
H error
Larger error at close approaches
t
Keplers third law is not respected
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Adaptive time steps are highly desirable for
accuracy and symmetry
But Adaptivity can destroy the symplectic
shadowing structure CalvoSanz-Serna Adaptive
methods may not be efficient as a splitting method
AIM To construct efficient, adaptive, symplectic
methods EASY which respect symmetries
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H error
t
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Hamiltonian ODE system
The Sundman transform introduces a continuous
adaptive time step.
IDEA Introduce a fictive computational time
SMALL if solution requires small time-steps
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Rescaled system for p,q and t
Can make Hamiltonian via the Poincare Transform
New variables
Hamiltonian
Now solve using a Symplectric ODE solver
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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.
Approach insist that the performance of the
numerical method when using the computational
variable should be independent of the scale of
the solution and that the method should respect
the symmetries of the ODE
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The differential equation system
Is invariant under scaling if it is unchanged by
the symmetry
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 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 symmetry
Numerical scheme is scale-invariant if
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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 change
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H Error
Surprisingly sharp!!!
Method order
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Scale invariant methods for PDES
These methods extend naturally to PDES with
scaling and other symmetries
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Examples
Parabolic blow-up High-order blow-up
NLS Chemotaxis
PME Rainfall
Need to continuously adapt in time and space
Introduce spatial analogue of the fictive time
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Adapt spatially by mapping a uniform mesh from a
computational domain into a physical domain
Use a strategy for computing the mesh which takes
symmetries into account
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Introduce a mesh potential
Geometric scaling
Control scaling via a measure
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Evolve mesh by solving a MK based PDE
(PMA)
Spatial smoothing (Invert operator using a
spectral method)
Ensures right-hand-side scales like P in
d-dimensions to give global existence
Averaged measure
Parabolic Monge-Ampere equation PMA
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Because PMA is based on a geometric approach, it
has natural symmetries
1. System is invariant under translations and
rotations
2. For appropriate choices of M the system is
invariant under scaling symmetries
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PMA is scale invariant provided that
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Example Parabolic blow-up in d dimensions
Scale
Regularise
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• Basic
  approach
• Discretise PDE and PMA in the computational
  domain
• Solve the coupled mesh and PDE system either
• (i) As one large system (stiff!)
  or
• (ii) By alternating between PDE and mesh

Method admits exact discrete self-similar
solutions
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solve PMA simultaneously with
the PDE
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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
Now extending it to CFD problems Eady, Bousinessq