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Takuya Tsuchiya

Publications and source records attributed to Takuya Tsuchiya.

At least 19 recordsLinked to original sources
arXiv2025

Structure-preserving numerical calculation of wave equation for a vector field

For the Proca equation, which is a wave equation for a vector field, we derive the canonical formulation including constraints from the Stueckelberg action and propose discrete equations with a structure-preserving scheme for conserving the constraints at the discrete level. Numerical simulations are performed using these discrete equations and other discrete equations with a standard scheme. We show the results obtained using the structure-preserving scheme and provide more accurate and stable numerical solutions.

math.NA↗
arXiv2024

Hadamard's variational formula for simple eigenvalues

We study Hadamard's variational formula for simple eigenvalues under dynamical and conformal deformations. Particularly, harmonic convexity of the first eigenvalue of the Laplacian under the mixed boundary condition is established for two-dimensional domain, which implies several new inequalities.

math.AP↗
arXiv2024

Stability analysis and improvement of the covariant BSSN formulation against the FLRW spacetime background

In this study, we investigate the numerical stability of the covariant Baumgarte--Shapiro--Shibata--Nakamura (cBSSN) formulation against the Friedmann--Lema\^itre--Robertson--Walker spacetime. To evaluate the numerical stability, we calculate the constraint amplification factor by the eigenvalue analysis of the evolution of the constraint. We propose a modification to the time evolution equations of the cBSSN formulation for a higher numerical stability. Furthermore, we perform numerical simulations using the modified formulation to confirm its improved stability.

gr-qc↗
arXiv2024

Hadamard variation of eigenvalues with respect to general domain perturbations

We study Hadamard variation of eigenvalues of Laplacian with respect to general domain perturbations. We show their existence up to the second order rigorously and characterize the derivatives, using associated eigenvalue problems in finite dimensional spaces. Then smooth rearrangement of multiple eigenvalues is explicitly given. This result follows from an abstract theory, applicable to general perturbations of symmetric bilinear forms.

math.SP↗
arXiv2024

Error estimation for finite element method on meshes that contain thin elements

In an error estimation of finite element solutions to the Poisson equation, we usually impose the shape regularity assumption on the meshes to be used. In this paper, we show that even if the shape regularity condition is violated, the standard error estimation can be obtained if "bad" elements elements that violate the shape regularity or maximum angle condition are covered virtually by simplices that satisfy the minimum angle condition. A numerical experiment illustrates the theoretical result.

math.NA↗
arXiv2023

Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption, Part 2: Lagrange interpolation on tetrahedrons

This is the second lecture note on the error analysis of interpolation on simplicial elements without the shape regularity assumption (the previous one is arXiv:1908.03894). In this manuscript, we explain the error analysis of Lagrange interpolation on (possibly anisotropic) tetrahedrons. The manuscript is not intended to be a research paper. We hope that, in the future, it will be merged into a textbook on the mathematical theory of the finite element methods.

math.NA↗
arXiv2023

Liouville's formulae and Hadamard variation with respect to general domain perturbations

We study Hadamard variations with respect to general domain perturbations, particularly for the Neumann boundary condition. They are derived from new Liouville's formulae concerning the transformation of volume and area integrals. Then, relations to several geometric quantities are discussed; differential forms and the second fundamental form on the boundary.

math.AP↗
arXiv2022

Numerical accuracy and stability of semilinear Klein--Gordon equation in de Sitter spacetime

Numerical simulations of the semilinear Klein--Gordon equation in the de Sitter spacetime are performed. We use two structure-preserving discrete forms of the Klein--Gordon equation. The disparity between the two forms is the discretization of the differential term. We show that one of the forms has higher numerical stability and second-order numerical accuracy with respect to the grid, and we explain the reason for the instability of the other form.

math.NA↗
arXiv2022

Anisotropic interpolation error estimates using a new geometric parameter

We present precise anisotropic interpolation error estimates for smooth functions using a new geometric parameter and derive inverse inequalities on anisotropic meshes. In our theory, the interpolation error is bounded in terms of the diameter of a simplex and the geometric parameter. Imposing additional assumptions makes it possible to obtain anisotropic error estimates. This paper also includes corrections to an error in Theorem 2 of our previous paper, "General theory of interpolation error estimates on anisotropic meshes" (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191).

math.NA↗
arXiv2022

Numerical simulations of semilinear Klein--Gordon equation in the de Sitter spacetime with structure-preserving scheme

We perform some simulations of the semilinear Klein--Gordon equation in the de Sitter spacetime. We reported the accurate numerical results of the equation with the structure-preserving scheme (SPS) in an earlier publication (Tsuchiya and Nakamura in J. Comput. Appl. Math. \textbf{361}: 396--412, 2019). To investigate the factors for the stability and accuracy of the numerical results with SPS, we perform some simulations with three discretized formulations. The first formulation is the discretized equations with SPS, the second one is with SPS that replaces the second-order difference as the standard second-order central difference, and the third one is with SPS that replaces the discretized nonlinear term as the standard discretized expression. As a result, the above two replacements in SPS are found to be effective for accurate simulations. On the other hand, the ingenuity of replacing the second-order difference in the first formulation is not effective for maintaining the stability of the simulations.

math.NA↗
arXiv2022

Stable numerical simulation of Einstein equations in gravitational collapse space--time

We perform simulations in a gravitational collapsing model using the Einstein equations. In this paper, we review the equations for constructing the initial values and the evolution form of the Einstein equations called the BSSN formulation. In addition, since we treat a nonvacuum case, we review the evolution equations of the matter fields of a perfect fluid. To make the simulations stable, we propose a modified system, which decreases numerical errors in analysis, and we actually perform stable simulations with decreased numerical errors.

gr-qc↗
arXiv2022

On the stability of covariant BSSN formulation

In this study, we investigate the numerical stability of the covariant BSSN (cBSSN) formulation proposed by Brown. We calculate the constraint amplification factor (CAF), which is an eigenvalue of the coefficient matrix of the evolution equations of the constraints on the cBSSN formulation and on some adjusted formulations with constraints added to the evolution equations. The adjusted formulations have a higher numerical stability than the cBSSN formulation from the viewpoint of the CAF.

gr-qc↗
arXiv2022

Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption Part 1: Lagrange interpolation on triangles

In the error analysis of finite element methods, the shape regularity assumption on triangulations is typically imposed to obtain a priori error estimations. In practical computations, however, very thin or degenerated elements that violate the shape regularity assumption may appear when we use adaptive mesh refinement. In this manuscript, we attempt to establish an error analysis approach without the shape regularity assumption on triangulations. We have presented several papers on the error analysis of finite element methods on non-shape regular triangulations. The main points in these papers are that, in the error estimates of finite element methods, the circumradius of the triangles is one of the most important factors. The purpose of this manuscript is to provide a simple and plain explanation of the results to researchers and, in particular, graduate students who are interested in the subject. Therefore, this manuscript is not intended to be a research paper. We hope that, in the future, it will be merged into a textbook on the mathematical theory of the finite element methods.

math.NA↗
arXiv2021

A new geometric condition equivalent to the maximum angle condition for tetrahedrons

For a tetrahedron, suppose that all internal angles of faces and all dihedral angles are less than a fixed constant $C$ that is smaller than $π$. Then, it is said to satisfy the maximum angle condition with the constant $C$. The maximum angle condition is important in the error analysis of Lagrange interpolation on tetrahedrons. This condition ensures that we can obtain an error estimation, even on certain kinds of anisotropic tetrahedrons. In this paper, using two quantities that represent the geometry of tetrahedrons, we present an equivalent geometric condition to the maximum angle condition for tetrahedrons.

math.NA↗
arXiv2021

General theory of interpolation error estimates on anisotropic meshes

We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents the flatness of a tetrahedron. Through the introduction of the geometric parameter, the error estimates newly obtained can be applied to cases that violate the maximum-angle condition.

math.NA↗
arXiv2021

A robust discontinuous Galerkin scheme on anisotropic meshes

Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.

math.NA↗
arXiv2020

Crouzeix-Raviart and Raviart-Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition

We investigate the piecewise linear nonconforming Crouzeix-Raviar and the lowest order Raviart-Thomas finite-element methods for the Poisson problem on three-dimensional anisotropic meshes. We first give error estimates of the Crouzeix-Raviart and the Raviart-Thomas finite-element approximate problems. We next present the equivalence between the Raviart-Thomas finite-element method and the enriched Crouzeix-Raviart finite-element method. We emphasise that we do not impose either shape-regular or maximum-angle condition during mesh partitioning. Numerical results confirm the results that we obtained.

math.NA↗