FoCM 2014 conference

Workshop B4 - Geometric Integration and Computational Mechanics

December 15, 15:30 ~ 15:55 - Room B11

## Discrete inequalities for central-difference type operators

### Takayasu Matsuo

### University of Tokyo, Japan - matsuo@mist.i.u-tokyo.ac.jp

One advantage of the energy-preserving methods is that sometimes the energy gives an a priori estimate for the (numerical) solution. For example, in the cubic nonlinear Schroedinger equation, the quartic energy function (the Hamiltonian) yields an estimate $\|u\|_{\infty}<\infty$ for all $t>0$, with the aid of the discrete Gagliardo--Nirenberg and Sobolev inequalities.

Although such discrete inequalities have been known for the simplest forward (i.e. one-sided) finite difference operator, it remained open for more general operators including the standard central-difference operator, as far as the authors know. Accordingly, the analyses of energy-preserving methods with such operators remained open as well.

Recently, we found a unified way of establishing discrete inequalities for a certain range of central-difference type operators. In this talk, we show some results, and illustrate them through applications to some structure-preserving numerical schemes.

Joint work with Daisuke Furihata (Osaka University) and Hiroki Kojima (University of Tokyo).