Workshop on Structure-Preserving Methods for PDEs

Workshop on
Structure-Preserving Methods
for Partial Differential Equations

March 17th. (Tue.) -- 19th. (Thu.), 2009.
At the University of Tokyo, Japan.

menu:	Aim of the workshop Workshop venue Speakers Program
	Accommodation Information for foreign visitors Contact address Sponsorship

ATTN!! Workshop poster (PDF, 1,533KB)

* Aim of the workshop
As the structure-preserving methods --- or the "geometric numerical integration" methods --- for ordinary differential equations have reached a certain maturity, such methods for partial differential equations have drawn more and more interest in recent years. This workshop is concerned with the recent progresses in structure-preserving methods for PDEs and related areas.

Possible workshop topics include

Methods preserving invariants (conservative methods)
Methods preserving dissipation properties
Multi-symplectic methods
Related geometric numerical integration methods for ODEs

Proposals of contributed talks are welcome; contact T. Matsuo.

* Workshop venue
The workshop will be held at:

Seminar room A (on the 3rd floor),
Faculty of Engineering Building 6,
Hongo campus,
The University of Tokyo.
Address: 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656 Japan.

Useful maps:

Faculty of Engineering Bulding 6 in Hongo campus map
Access to the Hongo campus

* Speakers
Plenary talks:

Chris J. Budd (University of Bath, UK)
Daisuke Furihata (Osaka University, Japan)
Jialin Hong (Chinese Academy of Sciences, China)
Yuji Ishimori (Toyama Prefectural University, Japan)
Brynjulf Owren (NTNU, Norway)

Contributed talks:

Ikuo Saitoh (Future University-Hakodate, Japan)
Keigo Nitadori (The University of Tokyo, Japan)
Takaharu Yaguchi (The University of Tokyo, Japan)
Takayasu Matsuo (The University of Tokyo, Japan)

* Program
Final version as of March 10th.
We are planning to hold a small party in the evening of 18th (Wed.).

Program table

	morning (10-12)	afternoon (14-17)
17 (Tue.)	---	Furihata Yaguchi Matsuo
18 (Wed.)	Budd Saitoh	Owren Hong
19 (Thu.)	Ishimori Nitadori	---

Program with titles and abstracts

17 (Tue.): The Discrete Variational Method and Its Recent Progresses
- 14:00-15:00 D. Furihata (Osaka University, Japan)
  1) "Introduction to the Discrete Variational Derivative Method"
  2) "Challenge for Multi-Dimensional Cases I: on Graphs and Voronoi Meshes"
  To apply the discrete variational derivative method for multi-dimensional cases with arbitrary meshes, the author has gone back to some problems on graphs and found that the DVDM is available for them under some basic conditions on difference operators and graphs. On Voronoi meshes, similar conditions are satisfied naturally and the DVDM is available. In this talk, those basic conditions for operators and some DVDM applications on graphs and Voronoi meshes are shown.
- 15:15-15:45 T. Yaguchi (The University of Tokyo, Japan)
  "Challenge for Multi-Dimensional Cases II: on Non-Uniform Meshes"
  In multi-dimensional problems the use of non-uniform meshes are of importance. In this talk a strategy to extend the discrete variational derivative method to such meshes is described. As a simple example an extension to logically rectangular meshes is provided. As a challenge for adaptive meshes a numerical computation of the KdV equation on a moving mesh is also shown.
- 16:00-16:30 T. Matsuo (The University of Tokyo, Japan)
  "Challenge for Multi-Dimensional Cases III: Galerkin Approach"
  As a challenge for multi-dimensional cases, a Galerkin (or finite-element) extension of the discrete variational derivative method has been proposed in 2008 by the present author, and its capability has been confirmed in several PDEs such as the Camassa--Holm type equations. In this talk, after a brief summary of those previous works, some recent results are shown, including numerical computation of the Ginzburg--Landau vortices in two-dimensional superconductivity problems.
18 (Wed.): New Trends in Structure-Preserving Methods for PDEs
- 10:00-10:45 I. Saitoh (Future University-Hakodate, Japan)
  "Symplectic FDTD Methods for Maxwell Equations - Formulation, Discretization and Application"
  Maxwell Equations are the cannonical equations for Hamiltonian of the electro-magnetic helicity functional. The Finite Difference Time Domain (FDTD) method is a symplectic method, we extend FDTD method to symplectic implicit methods or higher order methods.
- 11:00-12:00 C. J. Budd (University of Bath, UK)
  "Adaptive Moving Mesh GI Methods for PDES"
  Adaptivity is important in the numerical solution of many PDEs. Moving mesh based adaptive methods have the proprty that they can inherit many of the important qualitative features of a PDE and are thus genuine GI methods. In this talk I will construct a geometrically based moving mesh procedure based on ideas from optimal transport. I will then show how this can be coupled to a PDE through a semi-discretisation procedure and will that this method, when linked to a suitable (scale invariant) method for solving the ODEs resulting from the semi-discretisation is extremely effective at solving a large number of PDEs. An important reason for this effectiveness is the invariance of both the equations describing the moving mesh and the discretisation of the PDE on such a mesh, to the action of both Galilean and scaling symmetries. I will include examples from gas dynamics and meteorology.
  This is joint work with Emily Walsh (Bath) and JF Williams (SFU)
- 14:00-15:00 B. Owren (NTNU, Norway)
  "On Conservative, Linearly Implicit Difference Schemes for Hamiltonian PDEs"
  We consider a general technique for deriving semi-explicit, energy preserving finite difference schemes for an important class of nonlinear Hamiltonian PDEs. The schemes can be made time-symmetric and in many cases free parameters can be chosen to improve certain properties of the scheme, like e.g. stability. In particular, we shall discuss two well-known schemes found in the literature for solving the cubic Schrodinger equation (CSE). These schemes are both symmetric, semi-explicit and conservative, yet they exhibit a fundamentally different behavior due to their stability properties as seen in numerical experiments. A stability analysis based on plane wave solutions to the CSE confirms this behavior, and can furthermore be used together with the above derivation technique to stabilize the ´´bad´´ scheme while maintaining energy conservation.
  This is joint work with Morten Dahlby.
- 15:15-16:15 J. Hong (Chinese Academy of Sciences, China)
  "The Multi-Symplectic Methods for Hamiltonian PDEs"
  In this talk we give some new results on multi-symplectic methods for Hamiltonian PDEs, we also present some applications of multi-symplectic methods to some important models in physics.
19 (Thu.): Related New Topics in ODEs
- 10:00-11:00 Y. Ishimori (Toyama Prefectural University, Japan)
  "Parallel Composition Methods for Conservative Numerical Integrations of Hamiltonian Equations"
  Parallel composition method (PCM) is a method of constructing high-order one-step schemes by a parallel connection of the 2nd-order multi-stage schemes. If PCM is applied to Hamilton's equations of motion, it has an advantage of preserving conservative properties, especially energy conservation. In the talk, the theory and simple numerical examples of PCM are presented.
- 11:15-12:00 K. Nitadori (The University of Tokyo, Japan)
  "Recent Progresses on Time-Symmetric Numerical Methods in Astrophysical N-Body Simulations" I talk on some recent progress of time-symmetric methods for astrophysical gravitational N-body simulations.
  First I introduce Hermite predictor-corrector integrators which exploit direct calculations of time derivatives of gravitational force. The fourth-order Hermite integrator which employs the first derivative of acceleration (jerk) has been widely used. Recently, the sixth-order one using up to the second derivative (snap) and the eighth-order one using up to the third derivative (crackle) was developed (Nitadori & Makino 2008). All three integrators use only the informations of the starting point and the ending point of the integration for the corrector, and consequently, they are time-symmetric after a convergence of P(EC)^n iterations.
  In the case of constant timestep, it is simple to symmetrize. I demonstrate some calculations of Kepler problem with a small eccentricity. It is not simple for variable timestep schemes; one needs to determine the timestep not to brake the symmetry. Furthermore, in astrophysical N-body simulations, we use a scheme called individual timestep scheme which allow each particle to have its own timestep. It seems to be more complicated to symmetrize it. Makino et al. 2006 suggested a time-symmetric individual timestep algorithm in which trajectories of all the particle are memorized during a certain time and iterations are performed according to the trajectories. However, their implementation was with a second-order integrator and the number of particle was limited up to 512. I have developed a Hermite integrator version of this algorithm and show some results of numerical test.

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* Accommodation
There are many hotels in Tokyo, from moderate to luxury. One good choice is:

Forest Hongo Hotel,

which is at walking distance from the workshop venue (about 10 min.).
It is also possible to arrange moderate rooms inside the Hongo campus; those who hope to keep this should contact the address below.

* Information for foreign visitors
Please see Arriving at Japan and at the university of Tokyo page.

* Contact address
Takayasu MATSUO
Department of Mathematical Informatics
Graduate School of Information Science and Technology
The University of Tokyo
mail: matsuo(at)mist.i.u-tokyo.ac.jp
Address: 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656 Japan.
Phone/Fax: +81-3-5841-6911

* Sponsorship
The workshop is held under the auspices of the Global Center of Excellence "The research and training center for new development in mathematics".

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