Takaharu Yaguchi

An extension of the discrete variational method to nonuniform grids

The discrete variational method is a method used to derive finite difference schemes that inherit the conservation/dissipation property of the original equations. Although this method has mainly been developed for uniform grids, we extend this method to multidimensional nonuniform meshes.

Preserving multiple first integrals by discrete gradients

We consider systems of ordinary differential equations with known first integrals. The notion of a discrete tangent space is introduced as the orthogonal complement of an arbitrary set of discrete gradients. Integrators which exactly conserve all the first integrals simultaneously are then defined. In both cases we start from an arbitrary method of a prescribed order (say, a Runge–Kutta scheme) and modify it using two approaches: one is based on projection and the other on local coordinates. The methods are tested on the Kepler problem.

Conservative numerical schemes for the Ostrovsky equation

The Ostrovsky equation describes gravity waves under the influence of Coriolis force. It is known that solutions of this equation conserve the L^2 norm and an energy function that is determined non-locally. In this paper we propose four conservative numerical schemes for this equation: a finite difference scheme and a pseudospectral scheme that conserve the norm, and the same types of schemes that conserve the energy. A numerical comparison of these schemes is also provided, which indicates that the energy conservative schemes perform better than the norm conservative schemes.

Numerical integration of the Ostrovsky equation based on its geometric structures

We consider structure preserving numerical schemes for the Ostrovsky equation, which describes gravity waves under the influence of Coriolis force. This equation has two associated invariants: an energy function and the L^2 norm. It is widely accepted that structure preserving methods such as invariants-preserving and multi-symplectic integrators generally yield qualitatively better numerical results. In this paper we propose five geometric integrators for this equation: energy-preserving and norm-preserving finite difference and Galerkin schemes, and a multi-symplectic integrator based on a newly found multi-symplectic formulation. A numerical comparison of these schemes is provided, which indicates that the energy-preserving finite difference schemes are more advantageous than the other schemes.

A New Characteristic Nonreflecting Boundary Condition for the Multidimensional Navier-Stokes Equations

A characteristic nonreflecting boundary condition for the Navier-Stokes equations is described. Because the computational resources are finite, one needs to truncate the computational domain when he/she simulates a physical problem. This truncation gives rise to non-physical artificial boundaries and one cannot obtain proper solutions without appropriate boundary conditions on such boundaries. Practically nonreflecting boundary conditions, which are boundary conditions that prevent the generation of reflections, are of great importance. One of the most popular methods for the Navier-Stokes equations right now is the Pionsot‐Lele boundary condition. Their boundary condition is based on Thompson’s boundary condition for the Euler equations, which is, however, theoretically one-dimensional, and hence is valid only when the flow is perpendicular to the boundary. Here we propose a boundary condition for the Euler equations which does not have the assumption for the direction of the flow. We also discuss its extension to the Navier-Stokes equations. Our basic idea is to estimate the direction of the flow from numerical data. Assuming that there are only simple waves, we derived a form of the ordinary dierential system of the characteristic curves for the multi-dimensional Euler equations. By using this form we can apply flux vector splitting and calculate the velocity of each wave packet numerically.

Measurement and visualization of face‐to‐face interaction among community‐dwelling older adults using wearable sensors

A number of interventions have been undertaken to develop and promote social networks among community‐dwelling older adults. However, it has been difficult to examine the effects of these interventions, because of problems in assessing interactions. The present study was designed to quantitatively measure and visualize face‐to‐face interactions among elderly participants in an exercise program. We also examined relationships among interactional variables, personality and interest in community involvement, including interactions with the local community.

Voronoi Random Fields

We propose a random field regarded as a generalization of Voronoi diagrams for the case where the positions of generators are distributed probabilistically. Our approach is a kind of stochastic approaches to Voronoi diagrams; however our definition is different from usual random Voronoi diagrams such as Poisson Voronoi diagrams. As numerical examples, first we discuss the case where the positions of generators are mutually independent and their marginal distributions are uniform distributions on disks. Second, we discuss the case where the distributions are given in the form of digital pictures.

Invariance of Furihata’s discrete gradient schemes for the Webster equation with different Riemannian structures

We consider invariance of schemes derived by using the discrete gradient method for the Webster equation under change of Riemannian structures. In our previous research we expected that Furihata’s discrete gradient method for the Webster equation has invariance under change of Riemannian structures. In this paper we prove this conjecture.

Hamiltonian structures of wave-type equations compatible with the finite element exterior calculus

As it is widely accepted, for differential equations that reflect some physical properties it is preferable to use numerical schemes that inherit these properties. Many of such schemes are designed for Hamiltonian equations and are derived by using the Hamiltonian structures of the equations. In this paper, we formulate Hamiltonian structures for a class of wave-type equations that are compatible with the finite element exterior calculus. The finite element exterior calculus is a unified approach to designing finite element schemes for discretizing the scalar Laplacian and the vector Laplacian. In this theory, the stability result is obtained by using the Hodge theory and the Poincare inequality. We provide Hamiltonian structures for the wave-type equations for which the schemes derived with the help of the finite element exterior calculus can be employed and thereby make combinations of structure-preserving methods and the finite element exterior calculus possible.

Application of the Lagrangian approach of the discrete gradient method to scleronomic holonomic systems

The discrete gradient method is a method to obtain energy–preserving numerical schemes for Hamiltonian systems. Recently, a Lagrangian counter part of this method is developed by Yaguchi, whereby energy–preserving schemes for the Euler–Lagrange equation without constraints can be derived. In this work, we extend this method to systems with scleronomic holonomic constraints.