A truncation error estimate of the interpolant of a particle method based on the Voronoi decomposition

Imoto, Yusuke; Tagami, Daisuke

doi:10.14495/jsiaml.8.29

Cited by 6 publications

(11 citation statements)

References 7 publications

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“…Remark 3 An example of a weight function satisfying Hypothesis 1 with k 0 = 2 on d = 2 is shown in [4].…”

Section: Hypothesis 2 Set a Function W Defined Inmentioning

confidence: 99%

“…For the class of particle methods, there are just a few studies of numerical analysis; see, for example, Ishijima-Kimura [3] and references therein. In [4], we have introduced a particle method and have established a truncation error estimate of its interpolant. Hence, as the next step of the mathematical justification, we focus ourselves on truncation error estimates of approximate differential operators.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, under a sufficient condition for the weight function and a regularity of the family of discrete parameters as introduced in [4], we establish a truncation error estimate of an approximate gradient operator. Moreover, with an additional condition for the weight function, we establish a truncation error estimate of an approximate Laplace operator.…”

Section: Introductionmentioning

confidence: 99%

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Truncation error estimates of approximate differential operators of a particle method based on the Voronoi decomposition

Imoto

Tagami

2017

JSIAM Letters

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Truncation errors are considered for approximate differential operators with a class of particle methods. Introducing sufficient conditions for the weight function and a regularity of the family of discrete parameters leads to truncation error estimates of approximate gradient and Laplace operators with a particle method based on the Voronoi decomposition. Moreover, some numerical results agree well with theoretical ones.

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“…Remark 3 An example of a weight function satisfying Hypothesis 1 with k 0 = 2 on d = 2 is shown in [4].…”

Section: Hypothesis 2 Set a Function W Defined Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Truncation error estimates of approximate differential operators of a particle method based on the Voronoi decomposition

Imoto

Tagami

2017

JSIAM Letters

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“…In previous works, we established truncation error estimates for an interpolant, approximate gradient operator, and approximate Laplace operator of a generalized particle method in which the particle volumes were given as Voronoi volumes [11,12]. A generalized particle method is a numerical method that typically includes conventional particle methods, such as SPH and MPS.…”

Section: Introductionmentioning

confidence: 99%

Truncation error estimates of approximate operators in a generalized particle method

Imoto

2020

Japan J. Indust. Appl. Math.

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To facilitate the numerical analysis of particle methods, we derive truncation error estimates for the approximate operators in a generalized particle method. Here, a generalized particle method is defined as a meshfree numerical method that typically includes other conventional particle methods, such as smoothed particle hydrodynamics or moving particle semi-implicit methods. A new regularity of discrete parameters is proposed via two new indicators based on the Voronoi decomposition of the domain along with two hypotheses of reference weight functions. Then, truncation error estimates are derived for an interpolant, approximate gradient operator, and approximate Laplace operator in the generalized particle method. The convergence rates for these estimates are determined based on the frequency with which they appear in the regularity and hypotheses. Finally, the estimates are computed numerically and the results are shown to be in good agreement with the theoretical results.

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“…Moreover, to improve the accuracy of the explicit particle method, we consider an optimization of the discrete parameters based on the truncation error estimates of the generalized approximate operators [8,11,12]. In particular, defining the generalized approximate operators as a wider class of those used in particle methods enables us to consider an optimization of discrete parameters without imposed constraint conditions in each method.…”

Section: Introductionmentioning

confidence: 99%

Convergence study and optimal weight functions of an explicit particle method for the incompressible Navier–Stokes equations

2019

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To increase the reliability of simulations by particle methods for incompressible viscous flow problems, convergence studies and improvements of accuracy are considered for a fully explicit particle method for incompressible Navier-Stokes equations. The explicit particle method is based on a penalty problem, which converges theoretically to the incompressible Navier-Stokes equations, and is discretized in space by generalized approximate operators defined as a wider class of approximate operators than those of the smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) methods. By considering an analytical derivation of the explicit particle method and truncation error estimates of the generalized approximate operators, sufficient conditions of convergence are conjectured. Under these conditions, the convergence of the explicit particle method is confirmed by numerically comparing errors between exact and approximate solutions. Moreover, by focusing on the truncation errors of the generalized approximate operators, an optimal weight function is derived by reducing the truncation errors over general particle distributions. The effectiveness of the generalized approximate operators with the optimal weight functions is confirmed using numerical results of truncation errors and driven cavity flow. As an application for flow problems with free surface effects, the explicit particle method is applied to a dam break flow.

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A truncation error estimate of the interpolant of a particle method based on the Voronoi decomposition

Cited by 6 publications

References 7 publications

Truncation error estimates of approximate differential operators of a particle method based on the Voronoi decomposition

Truncation error estimates of approximate differential operators of a particle method based on the Voronoi decomposition

Truncation error estimates of approximate operators in a generalized particle method

Convergence study and optimal weight functions of an explicit particle method for the incompressible Navier–Stokes equations

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