2011 Tokyo Workshop onStructure-Preserving Methods
February 21st. (Mon.), 2011.
At the University of Tokyo, Japan. |

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The workshop will be held at:

Faculty ofUseful maps:Engineering Building 6,Room 270(which is on the 2nd 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

- 13:30--14:00
**Takayasu Matsuo**(Univ. of Tokyo, Japan)

*Recent progress of the discrete variational derivative method: a review*

The discrete variational derivative method is a method for designing geometric integrators for variational PDEs. It was originally proposed by Japanese researchers Furihata and Mori in the late 1990s, and then it has been intensively developed and studied by various authors. In this review, after briefly glancing the history, we give a short survey on the recent progress and the outlook.

- 14:15--14:45
**Daisuke Furihata**(Osaka University, Japan)

*Discrete Variational Derivative Method on Voronoi Mesh -- with an example for the Cahn--Hilliard equation --*

We have proposed a kind of structure preserving methods, discrete variational derivative method, to construct numerical schemes for PDEs. The key idea of this method is to discretize the variational structure of PDEs using rigorous definitions of discrete variational derivative. We cannot apply this method to problems on non-orthogonal meshes but the Voronoi mesh, it means that we can obtain some structure preserving schemes on arbitrary mesh points in higher dimension problems. For example, we apply this idea to the Cahn--Hilliard equation and will show those numerical solutions in the talk.

- 15:00--15:30
**Takaharu Yaguchi**(Univ. of Tokyo, Japan)

*On the Backward Error Analysis of the Discrete Variational Derivative Method*

In this talk, we apply the backward error analysis to some numerical schemes which are obtained by the discrete variational derivative method. It is shown that most of the schemes have a modified equation of the leading order with a variational structure. Additionally, we also consider existence of a solitary wave solution to the modified equation of the KdV equation. This is a joint work with C. Budd.

- 15:45--16:15
**Yuto Miyatake**(Univ. of Tokyo, Japan)

*Structure preserving finite difference schemes for the Ostrovsky equation*

We consider structure preserving integration of the Ostrovsky equation. This equation has two associated invariants, the norm and energy functions, and recently Yaguchi--Matsuo--Sugihara(2010) have proposed several finite difference schemes preserving the invariants.

In this talk, we first propose yet other conservative finite difference schemes, and confirm that they are more advantageous than the existing schemes. Next we find a multi-symplectic formulation of the equation, and derive multi-symplectic finite difference schemes based on the expression.

- 16:30--18:00
**[Main Lecture, UTNAS #16]**Erwan Faou (INRIA, France)

*Resonances in long time integration of the nonlinear Schroedinger equation*

In this talk, we will review some recent advances in long time simulation of Hamiltonian PDE, by focusing on the special case of the nonlinear Schroedinger equation with cubic nonlinearity. After discussing some results concerning the long time behavior of the exact solution (preservation of the actions, energy cascades), we will study the persistence of such qualitative behaviors by fully discrete splitting schemes. In particular, we will show how the choice of the number of grid points (a prime integer or not) or the stepsize (resonant or not) can lead to numerical instabilities, and on the other hand how implicit schemes are in general unable to reproduce correctly the energy exchanges. The main tool to analyze these phenomena is the use of backward error analysis for splitting methods under CFL applied to Hamiltonian PDEs, as stated in a recent common work with B. Grebert. We will detail this last result by showing how the numerical solution can be interpreted as the exact solution of a modified Hamiltonian PDE on which the resonance analysis can be performed as in the continuous case.

* This lecture is given as a part of UTNAS (University of Tokyo Numerical Analysis Seminar).

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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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