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Adaptivity and Symmetry for ODEs and PDEs

This talk explores the use of adaptive methods to solve ODEs and PDEs with structures on multiple scales, including uncoupled and coupled systems with symmetry. The focus is on variable step size adaptive methods for ODEs and scale invariant adaptive methods for PDEs. The talk presents examples such as the Kepler problem and discusses the need for adaptivity and the challenges it poses for preserving symmetry. The talk also introduces a new approach called EASY (Efficient Adaptive Symplectic Method) that aims to construct efficient and symplectic adaptive methods.

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Adaptivity and Symmetry for ODEs and PDEs

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  1. Adaptivity and symmetry for ODEs and PDEs Chris Budd

  2. Basic Philosophy ….. • ODES and PDEs develop structures • on manytime and length scales • Structures may beuncoupled (eg. Gravity waves and slow weather evolution) and needmulti-scale methods • Or they may becoupled, typically through (scaling) symmetries and can be resolved usingadaptive methods • Talk will look at • variable step size adaptive methods for ODES • scale invariant adaptive methods for PDES

  3. The need for adaptivity: the Kepler problem Conserved quantities: Hamiltonian Angular Momentum Symmetries: Rotation, Reflexion, Time reversal, Scaling Kepler's Third Law

  4. Kepler orbits Forward Euler Symplectic Euler Stormer Verlet

  5. FE Global error SV Main error H error Larger errorat close approaches t Kepler’s third lawis not respected

  6. Adaptive time steps are highly desirable foraccuracyandsymmetry But … Adaptivity can destroy the symplectic shadowing structure[Calvo+Sanz-Serna] Adaptive methods may not be efficient as a splitting method AIM: To construct efficient, adaptive, symplectic methods EASY which respect symmetries

  7. H error t

  8. Hamiltonian ODE system: The Sundman transformintroduces a continuous adaptive time step. IDEA: Introduce a fictive computational time SMALL if solution requires small time-steps

  9. Rescaled system forp,q and t Can make Hamiltonianvia thePoincare Transform New variables Hamiltonian Now solve using a Symplectric ODE solver

  10. Choice of the scaling functiong(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 themethod should respect the symmetries of the ODE

  11. The differential equation system Is invariant under scaling if it is unchanged by the symmetry eg. Kepler’s third law relating planetary orbits It generically admitsparticular self-similar solutionssatisfying

  12. 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

  13. Example: Kepler problem in radial coordinates A planet moving with angular momentum with radial coordinater = qand withdr/dt = psatisfies a Hamiltonian ODE with Hamiltonian If symmetry Numerical scheme is scale-invariant if

  14. If there are periodic solutions with close approaches Hard to integrate with a non-adaptive scheme q t

  15. Consider calculating them using the scaling No scaling Levi-Civita scaling Scale-invariant Constant angle change

  16. H Error Surprisingly sharp!!! Method order

  17. Scale invariant methods for PDES These methods extend naturally to PDES with scaling and other symmetries

  18. 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

  19. Adapt spatially by mapping a uniform mesh from a computational domain into a physicaldomain Use a strategy for computing the mesh which takes symmetries into account

  20. Introduce a mesh potential Geometric scaling Control scaling via a measure

  21. 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

  22. 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

  23. PMA is scale invariant provided that

  24. Example: Parabolic blow-up in d dimensions Scale: Regularise:

  25. 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

  26. solve PMAsimultaneously with the PDE 10 10^5 Solution: Y X Mesh:

  27. Solution in the computational domain 10^5 Same approach works well for the Chemotaxis eqns, Nonlinear Schrodinger eqn, Higher order PDEs Now extending it to CFD problems: Eady, Bousinessq

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