Yuto Miyatake

Conservative finite difference schemes for the Degasperis-Procesi equation

We consider the numerical integration of the Degasperis-Procesi equation, which was recently introduced as a completely integrable shallow water equation. For the equation, we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi-Hamiltonian form of the equation at the same time. We also prove the unique solvability of the schemes, and show some numerical examples.

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 Characterization of Energy-Preserving Methods and the Construction of Parallel Integrators for Hamiltonian Systems

High order energy-preserving methods for Hamiltonian systems are presented. For this aim, an energy-preserving condition of continuous stage Runge--Kutta methods is proved. Order conditions are simplified and parallelizable conditions are also given. The computational cost of our high order methods is comparable to that of the average vector field method of order two.

A derivation of energy-preserving exponentially-fitted integrators for Poisson systems

Abstract Exponentially-fitted (EF) methods are special methods for ordinary differential equations that better compute periodic/oscillatory solutions. Such solutions often appear in Hamiltonian systems, and in view of this, symplectic or energy-preserving variants of EF methods have been intensively studied recently. In these studies, the symplectic variants have been further applied to Poisson systems, while such a challenge has not ever been done for the energy-preserving variants. In this paper, we propose an energy-preserving EF method for Poisson systems, with special emphasis on the second- and fourth-order schemes.

A note on the adaptive conservative/dissipative discretization for evolutionary partial differential equations

An adaptive conservative or dissipative numerical method for nonlinear partial differential equations is established. The method not only inherits the conservation or dissipation property of the equation but also uses suitable non-uniform grids at each time step. Our numerical experiments indicate that the method is useful especially for localized solutions such as solitary wave solutions.

On a relationship between the T-congruence Sylvester equation and the Lyapunov equation

We consider the T-congruence Sylvester equation

On the equivalence between SOR-type methods for linear systems and the discrete gradient methods for gradient systems

AX+X^{\rm T}B=C

A fourth-order energy-preserving exponentially-fitted integrator for Poisson systems

, where

Solution of the k -th eigenvalue problem in large-scale electronic structure calculations

A\in \mathbb R^{m\times n}

Energy conservative/dissipative H1-Galerkin semi-discretizations for partial differential equations

,