2012 Tokyo Workshop onStructure-Preserving Methods
January 16th. (Mon.), 2012.
At the University of Tokyo, Japan. |

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The workshop will be held at:

Faculty ofUseful maps:Engineering Building 6,Room 367 (Seminar Room A)(3rd floor),

Hongo campus,

The University of Tokyo.

Address: 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656 Japan.

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

- 10:30--11:00
**Daisuke FURIHATA**(Osaka University, Japan)

*A simple guide to discrete variational derivative method and another attempt to overcome substantial computation cost of nonlinear problems*

We will show a simple and short introduction of discrete variational derivative method, which is one of structure preserving methods for partial differential equations, in the first part. Next we will show one of our attempts to overcome substantial cost to obtain numerical solutions for nonlinear problems. We already used to develop one method, which is based on decomposition of nonlinear polynomial terms to quadratic terms, for the difficulty, but this attempt is based on another simple idea and it is available to nonpolynomial nonlinear problems.

- 11:15--11:45
**Yoshio KOMORI**(Kyushu Institute of Technology, Japan)

*Weak second order S-ROCK methods for Stratonovich stochastic differential equations*

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In this talk our aim is to derive explicit Runge-Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations. This is a joint with Professor Kevin Burrage.

---Lunch break---

- 14:00--15:30
**[Main Lecture]**David Cohen (Basel University, Switzerland)

*HOP: From ODE to SPDE*

In the first part of the talk, we will consider (deterministic) Hamiltonian systems where high-frequency oscillations are generated by a linear part. For these kind of problems, we are interested in the long-time behaviour of the exact solution. Moreover, we will present a class of numerical methods, the trigonometric integrators, and explain their long-time behaviour. In the second part of the talk, we will discuss a generalisation of the trigonometric integrators to stiff second-order stochastic differential equations. Finally, numerical experiments for the stochastic wave equation will be presented.

**[Special session on integrable systems]**

- 16:00--16:30
**Yasuhiro OHTA**(Kobe University, Japan)

*Discretization of soliton equations derived through hodograph transformation*

Some soliton equations such as the Dym equation, short pulse equation and Camassa-Holm equation are derived from the Kadomtsev-Petviashvili hierarchy by applying the hodograph transformation. We discuss about the integrable discretization of these soliton equations. The keys of construction are bilinear form and determinant structure of solutions. Self-adaptive moving mesh schemes are obtained as discrete soliton equations in appropriate coordinate system.

- 16:45--17:15
**Takaharu YAGUCHI**(Kobe University, Japan)

*Backward Error Analysis of the Scheme for the KdV Equation by the Discrete Variational Derivative Method*

The discrete variational derivative method is a method for deriving numerical schemes that inherit the energy-conservation or -dissipation property. This method often derives gunnaturallyh good schemes despite the fact that it preserves "just one property." In order to investigate the reason why this method works so well, we apply the backward error analysis to the scheme for the KdV equation, as an example. It is shown by using the concentration-compactness method that a modified equation of the KdV equation admits a modified solitary wave solution. This is a joint work with C. Budd and D. Furihata.

- 17:30--18:00
**Daisuke TAKAHASHI**(Waseda University, Japan)

*Discrete and ultradiscrete mappings with an explicit Lyapunov function derived from the QRT system*

Quispel-Roberts-Thompson (QRT) system is a well-known integrable discrete mapping of second order. It has an explict conserved quantity and the solution is described by elliptic functions. We derive some versinos of discrete and ultradiscrete mappings extending the QRT mapping. They have an explicit Lyapunov function and an explicit attratctor. Continuous mappings are obtained from the discrete mappings through the continuum limit.

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Takayasu MATSUO

Department of Mathematical Informatics

Graduate School of Information Science and Technology

The University of Tokyo

mail:

Address: 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656 Japan.

Phone/Fax: +81-3-5841-6911

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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