arXiv subjects

Takuya Tsuchiya

Publications and source records attributed to Takuya Tsuchiya.

34 records · Page 2Linked to original sources

arXiv2019

Error Analysis of Lagrange Interpolation on Tetrahedrons

This paper describes the analysis of Lagrange interpolation errors on tetrahedrons. In many textbooks, the error analysis of Lagrange interpolation is conducted under geometric assumptions such as shape regularity or the (generalized) maximum angle condition. In this paper, we present a new estimation in which the error is bounded in terms of the diameter and projected circumradius of the tetrahedron. Because we do not impose any geometric restrictions on the tetrahedron itself, our error estimation may be applied to any tetrahedralizations of domains including very thin tetrahedrons.

math.NA↗

arXiv2019

On the numerical experiments of the Cauchy problem for semi-linear Klein-Gordon equations in the de Sitter spacetime

The computational analysis of the Cauchy problem for semi-linear Klein-Gordon equations in the de Sitter spacetime is considered. Several simulations are performed to show the time-global behaviors of the solutions of the equations in the spacetime based on the structure-preserving scheme. It is remarked that the sufficiently large Hubble constant yields the strong diffusion-effect which gives the long and stable simulations for the defocusing semi-linear terms. The reliability of the simulations is confirmed by the preservation of the numerically modified Hamiltonian of the equations.

gr-qc↗

arXiv2018

Error analysis of Crouzeix-Raviart and Raviart-Thomas finite element methods

We discuss the error analysis of the lowest degree Crouzeix-Raviart and Raviart-Thomas finite element methods applied to a two-dimensional Poisson equation. To obtain error estimations, we use the techniques developed by Babuška-Aziz and the authors. We present error estimates in terms of the circumradius and the diameter of triangles in which the constants are independent of the geometric properties of the triangulations. Numerical experiments confirm the results obtained.

math.NA↗

arXiv2018

Reorganizing topologies of Steiner trees to accelerate their elimination

We describe a technique to reorganize topologies of Steiner trees by exchanging neighbors of adjacent Steiner points. We explain how to use the systematic way of building trees, and therefore topologies, to find the correct topology after nodes have been exchanged. Topology reorganizations can be inserted into the enumeration scheme commonly used by exact algorithms for the Euclidean Steiner tree problem in $d$-space, providing a method of improvement different than the usual approaches. As an example, we show how topology reorganizations can be used to dynamically change the exploration of the usual branch-and-bound tree when two Steiner points collide during the optimization process. We also turn our attention to the erroneous use of a pre-optimization lower bound in the original algorithm and give an example to confirm its usage is incorrect. In order to provide numerical results on correct solutions, we use planar equilateral points to quickly compute this lower bound, even in dimensions higher than two. Finally, we describe planar twin trees, identical trees yielded by different topologies, whose generalization to higher dimensions could open a new way of building Steiner trees.

cs.DS↗

arXiv2018

Finite element approximations of minimal surfaces: algorithms and mesh refinement

Finite element approximations of minimal surface are not always precise. They can even sometimes completely collapse. In this paper, we provide a simple and inexpensive method, in terms of computational cost, to improve finite element approximations of minimal surfaces by local boundary mesh refinements. By highlighting the fact that a collapse is simply the limit case of a locally bad approximation, we show that our method can also be used to avoid the collapse of finite element approximations. We also extend the study of such approximations to partially free boundary problems and give a theorem for their convergence. Numerical examples showing improvements induced by the method are given throughout the paper.

math.NA↗

arXiv2017

The linear stability of the post-Newtonian triangular equilibrium in the three-body problem

Continuing work initiated in an earlier publication [Yamada, Tsuchiya, and Asada, Phys. Rev. D 91, 124016 (2015)], we reexamine the linear stability of the triangular solution in the relativistic three-body problem for general masses by the standard linear algebraic analysis. In this paper, we start with the Einstein-Infeld-Hoffman form of equations of motion for $N$-body systems in the uniformly rotating frame. As an extension of the previous work, we consider general perturbations to the equilibrium, i.e. we take account of perturbations orthogonal to the orbital plane, as well as perturbations lying on it. It is found that the orthogonal perturbations depend on each other by the first post-Newtonian (1PN) three-body interactions, though these are independent of the lying ones likewise the Newtonian case. We also show that the orthogonal perturbations do not affect the condition of stability. This is because these always precess with two frequency modes; the same with the orbital frequency and the slightly different one by the 1PN effect. The same condition of stability with the previous one, which is valid even for the general perturbations, is obtained from the lying perturbations.

gr-qc↗

arXiv2017

Constructing of constraint preserving scheme for Einstein equations

We propose a new numerical scheme of evolution for the Einstein equations using the discrete variational derivative method (DVDM). We derive the discrete evolution equation of the constraint using this scheme and show the constraint preserves in the discrete level. In addition, to confirm the numerical stability using this scheme, we perform some numerical simulations by discretized equations with the Crank-Nicolson scheme and with the new scheme, and we find that the new discretized equations have better stability than that of the Crank-Nicolson scheme.

gr-qc↗

arXiv2017

Approximating surface areas by interpolations on triangulations

We consider surface area approximations by Lagrange and Crouzeix--Raviart interpolations on triangulations. For Lagrange interpolation, we give an alternative proof for Young's classical result that claims the areas of inscribed polygonal surfaces converge to the area of the original surface under the maximum angle condition on the triangulation. For Crouzeix--Raviart interpolation we show that the approximated surface areas converge to the area of the original surface without any geometric conditions on the triangulation.

math.NA↗

arXiv2016

Constraint-Preserving Scheme for Maxwell's Equations

We derive the discretized Maxwell's equations using the discrete variational derivative method (DVDM), calculate the evolution equation of the constraint, and confirm that the equation is satisfied at the discrete level. Numerical simulations showed that the results obtained by the DVDM are superior to those obtained by the Crank-Nicolson scheme. In addition, we study the two types of the discretized Maxwell's equations by the DVDM and conclude that if the evolution equation of the constraint is not conserved at the discrete level, then the numerical results are also unstable.

gr-qc↗

arXiv2016

Extenting of Babuška-Aziz's theorem to higher-order Lagrange interpolation

We consider the error analysis of Lagrange interpolation on triangles and tetrahedrons. For Lagrange interpolation of order one, Babuška and Aziz showed that squeezing a right isosceles triangle perpendicularly does not deteriorate the optimal approximation order. We extend their technique and result to higher-order Lagrange interpolation on both triangles and tetrahedrons. To this end, we make use of difference quotients of functions with two or three variables. Then, the error estimates on squeezed triangles and tetrahedrons are proved by a method that is a straightforward extension of the original given by Babuška-Aziz.

math.NA↗

arXiv2015

A priori error estimates for Lagrange interpolation on triangles

We present the error analysis of Lagrange interpolation on triangles. A new \textit{a priori} error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are imposed in order to get this type of error estimates.

math.NA↗

arXiv2015

Post-Newtonian effects on the stability of the triangular solution in the three-body problem for general masses

Continuing work initiated in earlier publications [Ichita, Yamada and Asada, Phys. Rev. D {\bf 83}, 084026 (2011); Yamada and Asada, Phys. Rev. D {\bf 86}, 124029 (2012)], we examine the post-Newtonian (PN) effects on the stability of the triangular solution in the relativistic three-body problem for general masses. For three finite masses, a condition for stability of the triangular solution is obtained at the first post-Newtonian (1PN) order, and it recovers previous results for the PN restricted three-body problem when one mass goes to zero. The stability regions still exist even at the 1PN order, though the PN triangular configuration for general masses is less stable than the PN restricted three-body case as well as the Newtonian one.

gr-qc↗

arXiv2014

On the circumradius condition for piecewise linear triangular elements

We discuss the error analysis of linear interpolation on triangular elements. We claim that the circumradius condition is more essential then the well-known maximum angle condition for convergence of the finite element method. Numerical experiments show that this condition is the best possible. We also point out that the circumradius condition is closely related to the definition of surface area.

math.NA↗

arXiv2013

A Babuška-Aziz type proof of the circumradius condition

In this paper the error of polynomial interpolation of degree 1 on triangles is considered. The circumradius condition, which is more general than the maximum angle condition, is explained and proved by the technique given by Babuška-Aziz.

math.NA↗

arXiv2012

Constraint Propagation of $C^2$-adjusted Formulation II --- Another Recipe for Robust Baumgarte-Shapiro-Shibata-Nakamura Evolution System ---

In order to obtain an evolution system which is robust against the violation of constraints, we present a new set of evolution systems based on the so-called Baumgarte-Shapiro-Shibata-Nakamura (BSSN) equations.The idea is to add functional derivatives of the norm of constraints, $C^2$, to the evolution equations, which was proposed by Fiske (2004) and was applied to the ADM formulation in our previous study. We derive the constraint propagation equations, discuss the behavior of constraint damping, and present the results of numerical tests using the gauge-wave and polarized Gowdy wave spacetimes. The construction of the $C^2$-adjusted system is straightforward. However, in BSSN, there are two kinetic constraints and three algebraic constraints; thus, the definition of $C^2$ is a matter of concern. By analyzing constraint propagation equations, we conclude that $C^2$ should include all the constraints, which is also confirmed numerically. By tuning the parameters, the lifetime of the simulations can be increased as 2-10 times as longer than those of the standard BSSN evolutions.

gr-qc↗

arXiv2011

Constraint Propagation of $C^2$-adjusted Formulation - Another Recipe for Robust ADM Evolution System

With a purpose of constructing a robust evolution system against numerical instability for integrating the Einstein equations, we propose a new formulation by adjusting the ADM evolution equations with constraints. We apply an adjusting method proposed by Fiske (2004) which uses the norm of the constraints, C2. One of the advantages of this method is that the effective signature of adjusted terms (Lagrange multipliers) for constraint-damping evolution is pre-determined. We demonstrate this fact by showing the eigenvalues of constraint propagation equations. We also perform numerical tests of this adjusted evolution system using polarized Gowdy-wave propagation, which show robust evolutions against the violation of the constraints than that of the standard ADM formulation.

gr-qc↗