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Applying Geometric Integrators

24th April 2007 to 27th April 2007

Satellite Workshop held at ICMS, 14 India Street, Edinburgh

Organisers: Leimkuhler, Ben (University of Edinburgh),  McLachlan, Robert (Massey University)

Short Report

Geometric integrators are numerical discretization methods for solving time-dependent differential equations which exactly conserve phase-flow structures such as time-reversibility, symplectic structure, phase space volume, or first integrals. In recent years there has been an extraordinary interest in geometric integration methods among physicists and mathematicians with chemists and engineers also employing geometric integrators for many calculations. In many cases the benefits of using a geometric integration approach are so great compared to alternatives that no other technique can be envisioned; for example the use of symplectic methods in molecular dynamics appears to offer such an overwhelming advantage. In other large scale applications, some of the available geometric structure of a given problem must be sacrificed due to the severe practical constraints of computational efficiency, or choices must be made between different types of structures which can be individually, but not simultaneously, preserved. While much research on geometric integrators is motivated by applications, there is often a gulf between the developers of mathematical methods and the large body of applications scientists who stand to benefit the most from that work. This 3-day workshop brought together experts in geometric integration, primarily from the mathematics and physics communities, with the goal of highlighting barriers to the implementation and use of geometric integration methods, clarifying the relative importance of different types of structures in the practical setting, and considering possible new types of geometric integration methodology which may have a strong impact on modern computational science.

This meeting was a satellite event of the Isaac Newton Institute programme on Highly Oscillatory Problems, held at the head-quarters of ICMS at 14 India Street. This house is the birthplace of James Clerk Maxwell and is situated in the historic New Town of Edinburgh, very near the centre.


Tuesday 24 April
12.00-14.30 Registration and sandwich lunch
14.30-15.00 Hairer, Ernst (University of Geneva)
Modified differential equations
15.00-15.30 Bond, Stephen (University of Illinois)
On the accuracy of numerically computed averages in molecular simulation
15.30-16.00 Coffee break
16.00-16.30 Faou, Erwan (INRIA Rennes)
Molecular dynamics in a shaker
16.30-17.00 Legoll, Frederic (École Nationale des Ponts et Chaussées)
Derivation of symplectic numerical schemes for highly oscillatory Hamiltonian systems: a new approach?
17.00-17.10 Short break
17.10-17.40 Melnik, Roderick (WLU/Waterloo, Canada)
Conservative numerical approximations and the Cayley transform technique in applications to PDEs
19.30 Group meal at Nargile, Hanover Street
Wednesday 25 April
09:00-09:30 Skeel, Bob (Purdue University)
What makes molecular dynamics work
09:30-10:00 Sanz-Serna, Jesus (University of Valladolid)
Mollified impulse methods revisited
10.00-10.30 Coffee break
10.30-11.00 Tuckerman, Mark (New York University)
Statistical mechanical concepts and measure-preserving integrators for non-Hamiltonian systems
11.00-11.10 Short break
11.10-11.40 Izaguirre, Jesus (University of Notre Dame)
Longer time step integrator using normal-mode constraints
11.40-12.10 Cohen, David (Norwegian University of Science & Technology)
Highly oscillatory Hamiltonian systems with non-constant mass matrix
12.10-14.00 Lunch break
14.00-14.30 Quispel, Reinout (La Trobe University)
Integral-preserving integrators and linearization-preserving integrators
14.30-15.00 Budd, Chris (University of Bath)
Geometric integration and 2D mesh generation
15.00-15.30 Blanes Sergio (Universidad Politécnica de Valencia)
Adaptive splitting integrators
15.30 Go to King's Buildings for Wanner's colloquium
Thursday 26 April
09.00-09.30 Laskar, Jacques (IMCCE, Observatory of Paris)
Long time integrations of the Solar System
09.30-10.00 Lubich, Christian (University of Tübingen)
Symplectic integrators for numerical general relativity
10.00-10.30 Coffee break
10.30-11.00 Mikkola, Seppo (Turku Observatory)
Algorithmic regularization
11.00-11.10 Short break
11.10-11.40 Stadel, Joachim (University of Zurich)
Multi-stepping integrators in astrophysical N-body system
11.40-12.10 Kennedy, Anthony (University of Edinburgh)
12.10-14.00 Lunch break
14.00-14.30 Grimm, Volker (Heinrich-Heine-Universität)
On the computation of cosine of operators in second order abstract differential equations
14.30-15.00 McLachlan, Robert (Massey University)
Multisymplectic integrators
15.00-15.30 Coffee break
15.30-16.00 Celledoni, Elena (NTNU)
Simulation of rigid body dynamics using Jacobi elliptic functions and integrals
16.00-16.30 Wulff, Caudia (University of Surrey)
Approximate energy conservation for symplectic time-semidiscretizations of semilinear wave equations
16.30-16.40 Short break
16.40-17.10 Dixon, Matthew (Imperial College London)
Moser-Veselov integrators for continuum dynamics
17.10-17.40 Zanna, Antonella (University of Bergen)
On the exact integration of Rigid Body equations
19.30 Workshop Dinner at Howies, Waterloo Place
Friday 27 April
09.00-09.30 Frank, Jason (CWI Amsterdam)
Statistical mechanics of Arakawa's Jacobian: numerical experiments
09.30-10.00 Owren, Brynjulf (NTNU Trondheim)
Geometric integrators for the Camassa-Holm equations
10.00-10.30 Coffee break
10.30-11.00 Fasso, Francesco (University of Padua)
The conjugate locus for the Euler top
11.00-11.10 Short break
11.10-11.40 Cotter, Colin (Imperial College London)
Particle-mesh methods for matching shapes
11.40-12.10 Casas Perez, Fernando (Universitat Jaume I)
On the convergence of Magnus and related expansions
12.10-14.00 Lunch break
14.00-14.30 Malham, Simon (Heriot-Watt University)
Lie group stochastic integrators
14.30-15.00 Leimkuhler, Ben (University of Edinburgh)
A generalized molecular dynamics model
15.00 End of workshop



Name Institution
Abdulle, Assyr University of Edinburgh
Blanes, Sergio Universidad Politécnica de Valencia
Bond, Stephen University of Illinois
Budd, Chris University of Bath
Casas Perez, Fernando Universitat Jaume I
Celledoni, Elena NTNU
Cohen, David Norwegian University of Science & Technology
Cotter, Colin Imperial College London
Dehnen, Walter University of Leicester
Dixon, Matthew Imperial College London
Faou, Erwan INRIA Rennes
Fasso, Francesco University of Padua
Frank, Jason CWI Amsterdam
Grimm, Volker Heinrich-Heine-Universität
Hairer, Ernst University of Geneva
Heggie, Douglas University of Edinburgh
Izaguirre, Jesus University of Notre Dame
Kennedy, Anthony University of Edinburgh
Laskar, Jacques IMCCE, Observatory of Paris
Legoll, Frederic École Nationale des Ponts et Chaussées
Leimkuhler, Ben University of Edinburgh
Lubich, Christian University of Tübingen
Malham, Simon Heriot-Watt University
McLachlan, Robert Massey University
Melnik, Roderick WLU/Waterloo, Canada
Mikkola, Seppo Turku Observatory
Noorizadeh, Emad University of Edinburgh
Owren, Brynjulf NTNU Trondheim
Quispel, Reinout La Trobe University
Sanz-Serna, Jesus University of Valladolid
Skeel, Bob Purdue University
Stadel, Joachim University of Zurich
Tuckerman, Mark New York University
Vanneste, Jacques University of Edinburgh
Wanner, Gerhard University of Geneva
Wulff, Claudia University of Surrey
Zanna, Antonella University of Bergen


University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons