Takayasu Matsuo

Discrete variational derivative method : a structure-preserving numerical method for partial differential equations

Preface Introduction and Summary of This Book An Introductory Example: the Spinodal Decomposition History Derivation of Dissipative or Conservative Schemes Advanced Topics Target Partial Differential Equations Variational Derivatives First-Order Real-Valued PDEs First-Order Complex-Valued PDEs Systems of First-Order PDEs Second-Order PDEs Discrete Variational Derivative Method Discrete Symbols and Formulas Procedure for First-Order Real-Valued PDEs Procedure for First-Order Complex-Valued PDEs Procedure for Systems of First-Order PDEs Design of Schemes Procedure for Second-Order PDEs Preliminaries on Discrete Functional Analysis Applications Target PDEs Cahn-Hilliard Equation Allen-Cahn Equation Fisher-Kolmogorov Equation Target PDEs Target PDEs Target PDEs Nonlinear Schrodinger Equation Target PDEs Zakharov Equations Target PDEs Other Equations Advanced Topic I: Design of High-Order Schemes Orders of Accuracy of the Schemes Spatially High-Order Schemes Temporally High-Order Schemes: With the Composition Method Temporally High-Order Schemes: With High-Order Discrete Variational Derivatives Advanced Topic II: Design of Linearly-Implicit Schemes Basic Idea for Constructing Linearly-Implicit Schemes Multiple-Points Discrete Variational Derivative Design of Schemes Applications Remark on the Stability of Linearly-Implicit Schemes Advanced Topic III: Further Remarks Solving System of Nonlinear Equations Switch to Galerkin Framework Extension to Non-Rectangular Meshes on D Region A Semi-discrete schemes in space B Proof of Proposition 3.4 Bibliography Index

Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind

In this paper we propose new numerical methods for linear Fredholm integral equations of the second kind with weakly singular kernels. The methods are developed by means of the Sinc approximation with smoothing transformations, which is an effective technique against the singularities of the equations. Numerical examples show that the methods achieve exponential convergence, and in this sense the methods improve conventional results where only polynomial convergence have been reported so far.

Spatially Accurate Dissipative or Conservative Finite Difference Schemes Derived by the Discrete Variational Method

A method, called “the discrete variational method”, has been recently presented by Furihata and Matsuo for designing finite difference schemes that inherit energy dissipation or conservation property from nonlinear partial differential equations (PDEs). In this paper the method is enhanced so that the derived schemes should be highly accurate in space by introducing higher order spatial difference operators, including the so-called “spectral differentiation” operator. Applications to the KdV equation, and the cubic nonlinear Schrödinger equation are also presented.

A stable, convergent, conservative and linear finite difference scheme for the Cahn-Hilliard equation

We propose a stable, convergent, conservative and linear finite difference scheme to solve numerically the Cahn-Hilliard equation. The proposed scheme realizes both linearity and stability. We show uniqueness, existence and convergence of the solution to the scheme. Numerical examples demonstrate the effectiveness of the proposed scheme.

Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

Error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation. Although in the literature various error estimates have already been given for these approximations, those estimates were basically for examining the rates of convergence, and several constants were left unevaluated. Giving more explicit estimates, i.e., evaluating these constants, is of great practical importance, since by this means we can reinforce the useful formulas with the concept of “verified numerical computations.” In this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems: the function to be approximated is analytic in a suitable region. We also improve some formulas themselves to decrease their computational costs. Numerical examples that confirm the theory are also given.

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.

High-order schemes for conservative or dissipative systems

We propose a new method for designing high-order finite difference schemes that inherit conservation or dissipation properties from conservative or dissipative systems such as Hamiltonian systems with/without damping terms. The proposed method has a feature that the computational costs of the resulting schemes do not increase in practice, even when the order of accuracy is increased.

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.

A Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation

A new Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation is presented. The scheme has an additional welcome feature that in exact arithmetic it is unconditionally stable in the sense that the solution is always bounded. Numerical examples that confirm the theory and the effectiveness of the scheme are also given.

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.