Atsushi Nobe

NUMERICAL ANALYSIS OF BREAKING WAVES USING THE MOVING PARTICLE SEMI-IMPLICIT METHOD

SUMMARY The numerical method used in this study is the moving particle semi-implicit (MPS) method, which is based on particles and their interactions. The particle number density is implicitly required to be constant to satisfy incompressibility. A semi-implicit algorithm is used for two-dimensional incompressible non-viscous flow analysis. The particles whose particle number densities are below a set point are considered as on the free surface. Grids are not necessary in any calculation steps. It is estimated that most of computation time is used in generation of the list of neighboring particles in a large problem. An algorithm to enhance the computation speed is proposed. The MPS method is applied to numerical simulation of breaking waves on slopes. Two types of breaking waves, plunging and spilling breakers, are observed in the calculation results. The breaker types are classified by using the minimum angular momentum at the wave front. The surf similarity parameter which separates the types agrees well with references. Breaking waves are also calculated with a passively moving float which is modelled by particles. Artificial friction due to the disturbed motion of particles causes errors in the flow velocity distribution which is shown in comparison with the theoretical solution of a cnoidal wave.

Ultradiscrete QRT maps and tropical elliptic curves

It is shown that the polygonal invariant curve of the ultradiscrete QRT (uQRT) map, which is a two-dimensional piecewise linear integrable map, is the complement of the tentacles of a tropical elliptic curve on which the curve has a group structure in analogy to classical elliptic curves. Through the addition formula of a tropical elliptic curve, a tropical geometric description of the uQRT map is then presented. This is a natural tropicalization of the geometry of the QRT map found by Tsuda. Moreover, the uQRT map is linearized on the tropical Jacobian of the corresponding tropical elliptic curve in terms of the Abel-Jacobi map. Finally, a formula concerning the period of a point in the uQRT map is given, and an exact solution to its initial-value problem is constructed by using the ultradiscrete elliptic theta function.

Exact solutions for discrete and ultradiscrete modified KdV equations and their relation to box-ball systems

A new class of solutions is proposed for discrete and ultradiscrete modified KdV equations. These are directly related to solutions of the box and ball system with a carrier. Moreover, an extended box and ball system and its exact solutions are discussed.

On reversibility of cellular automata with periodic boundary conditions

Reversibility of one-dimensional cellular automata with periodic boundary conditions is discussed. It is shown that there exist exactly 16 reversible elementary cellular automaton rules for infinitely many cell sizes by means of a correspondence between elementary cellular automaton and the de Bruijn graph. In addition, a sufficient condition for reversibility of three-valued and two-neighbour cellular automaton is given.

Ultradiscretization of solvable one-dimensional chaotic maps

We consider the ultradiscretization of a solvable one-dimensional chaotic map which arises from the duplication formula of the elliptic functions. It is shown that the ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously. A geometric interpretation of the dynamics of the tent map is given in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding to the mth multiplication formula of the elliptic functions is also discussed.

From cellular automaton to difference equation: a general transformation method which preserves time evolution patterns

We propose a general method to construct a partial difference equation which preserves any time evolution patterns of a cellular automaton. The method is based on inverse ultradiscretization with filter functions.

Ultradiscretization of elliptic functions and its applications to integrable systems

It is shown that there exist three kinds of ultradiscrete analogues of Jacobis elliptic functions. In this process, the asymptotic behaviour of the poles and the zeros of the functions plays a crucial role. Using the ultradiscrete analogues and an addition formula, exact solutions to the ultradiscrete KP equation are constructed, and their relation to the ultradiscrete QRT system is discussed.

Linearizable cellular automata

The initial value problem for a class of reversible elementary cellular automata with periodic boundaries is reduced to an initial-boundary value problem for a class of linear systems on a finite commutative ring . Moreover, a family of such linearizable cellular automata is given.

Stable Difference Equations Associated with Elementary Cellular Automata

We construct a difference equation which preserves any time evolution pattern of the rule 90 elementary cellular automaton. We also demonstrate that such difference equations can be obtained for any elementary cellular automata.

Periodic multiwave solutions to the Toda-type cellular automaton

An ultradiscretization of the Riemann theta function is proposed. The ultradiscretization satisfies an addition formula, which is an ultradiscrete analogue of an addition formula for the Riemann theta function. Using the addition formula, periodic multiwave solutions to the Toda-type cellular automaton are obtained.