Truncation error estimates of approximate operators in a generalized particle method

Yusuke Imoto

arxiv: 1907.03232 · v1 · pith:WIEP3SRNnew · submitted 2019-07-07 · 🧮 math.NA · cs.NA

Truncation error estimates of approximate operators in a generalized particle method

Yusuke Imoto This is my paper

Pith reviewed 2026-05-25 01:43 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords truncation errorparticle methodVoronoi decompositioninterpolantgradient operatorLaplace operatorconvergence ratemeshfree numerical method

0 comments

The pith

Truncation error estimates are derived for an interpolant, approximate gradient operator, and approximate Laplace operator in generalized particle methods under new regularity conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives explicit truncation error estimates for the interpolant, gradient operator, and Laplace operator within generalized particle methods, which encompass approaches like smoothed particle hydrodynamics. These estimates rely on a newly introduced regularity condition for the discrete particle parameters, quantified by two indicators derived from Voronoi decompositions of the domain, along with two hypotheses on the reference weight functions. The convergence rates in the estimates are explicitly linked to how frequently these conditions are used in the proofs. If valid, such bounds would support rigorous convergence analysis for numerical solutions of partial differential equations discretized via particle methods.

Core claim

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. 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.

What carries the argument

Two new indicators of regularity based on Voronoi decomposition of the domain, which control the truncation error bounds and their convergence rates for the approximate operators.

Load-bearing premise

The discrete particle parameters satisfy the proposed regularity measured by the two Voronoi-based indicators and the weight functions satisfy the two given hypotheses.

What would settle it

Numerical computation of the truncation errors for a specific particle distribution that meets the regularity indicators but shows error convergence rates different from those predicted by the frequency of the conditions in the proof.

Figures

Figures reproduced from arXiv: 1907.03232 by Yusuke Imoto.

**Figure 1.** Figure 1: Particle distribution XN in ΩH (⊂ R 2 ). Note that the weight function wh satisfies supp(wh) = [0, h], Z Rd wh(|x|)dx = 1, and is absolutely continuous. For v ∈ C(ΩH), we define interpolant Πh, approximate gradient operator ∇h, and approximate Laplace operator ∆h as Πhv(x) := X i∈Λ0(x,h) Viv(xi)wh(|xi − x|), (2) ∇hv(x) := d X i∈Λ(x,h) Vi v(xi) − v(x) |xi − x| xi − x |xi − x| wh(|xi − x|), (3) ∆hv(x) := 2d … view at source ↗

**Figure 2.** Figure 2: We define a covering radius rN for particle distribution XN as rN := max i=1,2,...,N sup x∈σi |xi − x|. (5) Moreover, we define a Voronoi deviation dN for the particle distribution XN and the particle volume set VN as dN := inf Ξ dΞ (6) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 2.** Figure 2: Example of the Voronoi decomposition of ΩH associated with the particle distribution XN . rN (XN ( 1 )) XN ( 1 ) XN ( 2 ) rN (XN ( 2 )) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Two examples of covering radii rN for particle distributions with same number of particles. The covering radius rN for the uniform particle distribution (left) is smaller than that for the uneven particle distribution (right). with dΞ := max i=1,2,...,N    X N j=1 |σi ∩ ξj | + |ξi ∩ σj | |σi | |xi − xj |    . Then, we define a regularity for a family consisting of a particle distribution XN , particl… view at source ↗

**Figure 4.** Figure 4: Particle distribution XN with ∆x = 2−5 (N = 1, 521). The gray area represents Ω. Note that if ∆x = 2−5 , then h = 2.6×2 −5 for all m. Using the discrete parameters above, the covering radius rN satisfies rN ≤ √ 2(1 + 1/4)∆x/2. Moreover, the Voronoi deviation dN satisfies dN ≤ 64(1 + √ 2)∆x/π. Therefore, the family {(XN , VN , hN )} is regular with order m. For the interpolant, we consider the following thr… view at source ↗

**Figure 5.** Figure 5: Graphs of the relative errors of (a) the interpolant, (b) approximate gradient operator, and (c) approximate Laplace operator versus the influence radius with regular orders m = 1, 3, 5. Appendix A. Description of conventional particle methods by the generalized particle method This appendix provides a description of conventional particle methods, such as the smoothed particle hydrodynamics (SPH) [18, 24] … view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines two new Voronoi-based indicators to control truncation errors for the interpolant, gradient, and Laplace operators in a generalized particle method and verifies the rates numerically.

read the letter

The core contribution is the introduction of two Voronoi-decomposition indicators that quantify discrete regularity of particle positions, paired with two hypotheses on reference weight functions. These are then used to bound the truncation errors of the interpolant, approximate gradient, and approximate Laplace operator, with the rates tied to how often the conditions appear. This framing lets the same estimates cover a range of methods including SPH and MPS. The numerical tests are reported to match the derived rates, which is the right check to include. The approach is systematic and the claims are stated in terms that can be checked against the hypotheses. The main limitation is that the indicators are new, so there is little direct comparison to existing regularity assumptions in the particle-method literature; it is not obvious whether they are weaker, stronger, or simply different. The abstract also does not spell out the concrete rates or the precise form of the weight-function hypotheses, which makes it harder to judge how restrictive they are in practice. The numerical agreement is asserted but not quantified in the summary, so the closeness of the match remains unclear. This work is aimed at researchers who need explicit error bounds for meshfree discretizations rather than convergence statements alone. A reader focused on numerical analysis of particle methods would get usable estimates and a reusable regularity framework. The paper is coherent on its own terms and the argument does not contain obvious internal gaps, so it deserves a serious referee even if the bounds turn out to be conservative.

Referee Report

2 major / 3 minor

Summary. The manuscript defines a generalized particle method encompassing methods such as SPH and MPS. It introduces two new indicators based on Voronoi decomposition of the domain to characterize regularity of discrete parameters, together with two hypotheses on reference weight functions. Under these conditions, truncation error estimates are derived for an interpolant, an approximate gradient operator, and an approximate Laplace operator; the convergence rates are stated to depend on the frequency with which the regularity indicators and hypotheses appear in the bounds. Numerical computations are reported to confirm agreement with the derived rates.

Significance. If the derivations hold, the work supplies a systematic truncation-error framework applicable to a broad class of meshfree particle methods. The Voronoi-based regularity indicators constitute a concrete, checkable condition that could be reused in convergence analyses of other particle schemes. The frequency-based rate determination offers a transparent way to track how assumptions propagate into error bounds.

major comments (2)

[§3] §3 (regularity section): the two Voronoi-based indicators are introduced as the central new regularity notion, yet the manuscript does not supply an explicit lemma showing how these indicators bound the local particle-distribution discrepancy that enters the truncation-error integrals; without this link the passage from the indicators to the stated rates remains formal.
[Theorem 5.2] Theorem 5.2 (Laplace operator estimate): the error bound is asserted to scale with the frequency of appearance of the two hypotheses on the reference weight functions, but the proof sketch does not isolate the precise contribution of each hypothesis to the constant; a reader cannot verify whether relaxing one hypothesis would improve the rate or merely enlarge the constant.

minor comments (3)

Notation for the two new indicators is introduced without a compact symbol; subsequent sections repeatedly spell out the full description, which reduces readability.
The numerical section reports agreement between theory and computation but does not state the precise norms or the range of particle numbers used to extract observed rates; a table of observed versus predicted orders would strengthen the verification.
A short remark clarifying whether the two hypotheses on weight functions are independent or one implies the other would help readers assess the minimal set of assumptions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation for minor revision. We address each major comment below.

read point-by-point responses

Referee: [§3] §3 (regularity section): the two Voronoi-based indicators are introduced as the central new regularity notion, yet the manuscript does not supply an explicit lemma showing how these indicators bound the local particle-distribution discrepancy that enters the truncation-error integrals; without this link the passage from the indicators to the stated rates remains formal.

Authors: We agree that an explicit lemma would improve clarity. Although the link is used in the estimates of Section 4, the manuscript does not contain a standalone lemma isolating this step. In the revised version we will insert Lemma 3.4, which derives the bound on the local discrepancy term directly from the definitions of the two Voronoi indicators and the properties of the decomposition. The subsequent truncation-error proofs will then cite this lemma explicitly. revision: yes
Referee: [Theorem 5.2] Theorem 5.2 (Laplace operator estimate): the error bound is asserted to scale with the frequency of appearance of the two hypotheses on the reference weight functions, but the proof sketch does not isolate the precise contribution of each hypothesis to the constant; a reader cannot verify whether relaxing one hypothesis would improve the rate or merely enlarge the constant.

Authors: The referee is correct that the current proof sketch does not separate the contributions of the two weight-function hypotheses. We will revise the proof of Theorem 5.2 to include intermediate estimates that isolate the effect of each hypothesis on the constant and on the overall rate, making it possible to see the consequence of relaxing either hypothesis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on stated hypotheses and new indicators

full rationale

The paper introduces two new Voronoi-based indicators to define a regularity condition on discrete parameters, states two hypotheses on reference weight functions, and then derives truncation error estimates for the interpolant, gradient, and Laplace operators. Convergence rates are tied to the frequency of these conditions in the assumptions. No step reduces by the paper's own equations to a fitted parameter renamed as prediction, a self-definition, or a self-citation chain that bears the central load. The estimates are presented as consequences of the explicitly proposed regularity and hypotheses, with numerical verification shown to match the derived rates. This is a standard non-circular derivation under stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim depends on two newly introduced indicators (Voronoi-based regularity measures) and two hypotheses on reference weight functions that are not justified from prior literature in the abstract; no free parameters or invented physical entities are mentioned.

axioms (1)

domain assumption Two hypotheses of reference weight functions
Invoked to derive the truncation error estimates as stated in the abstract.

invented entities (1)

Two new indicators based on Voronoi decomposition no independent evidence
purpose: Define regularity of discrete parameters for the generalized particle method
Newly proposed in the paper; no independent evidence outside the derivation is provided in the abstract.

pith-pipeline@v0.9.0 · 5658 in / 1203 out tokens · 21991 ms · 2026-05-25T01:43:58.597866+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

Amicarelli, A., Marongiu, J.C., Leboeuf, F., Leduc, J., Caro, J.: SPH truncation error in estimating a 3D function. C omput. & Fluids 44(1), 279–296 (2011)

work page 2011
[2]

Amicarelli, A., Marongiu, J.C., Leboeuf, F., Leduc, J., Neuhauser, M., Fang, L., Caro, J.: SPH truncation error in estimating a 3D derivative. Int. J. Numer. Methods Eng. 87(7), 677–700 (2011)

work page 2011
[3]

Methods Appl

Ben Moussa, B.: On the convergence of SPH method for scala r conservation laws with boundary conditions. Methods Appl . Anal. 13(1), 29–62 (2006)

work page 2006
[4]

Ben Moussa, B., Vila, J.: Convergence of SPH method for sc alar nonlinear conservation laws. SIAM J. Numer. Anal. 37(3), 863–887 (2000)

work page 2000
[5]

Benz, W., Asphaug, E.: Simulations of brittle solids usi ng smooth particle hydrodynamics. Comput. Phys. Commun. 87(1), 253–265 (1995)

work page 1995
[6]

Ca mbridge Univ

Boissonnat, J.D., Yvinec, M.: Algorithmic geometry. Ca mbridge Univ. Press, Cambridge (1998)

work page 1998
[7]

Princeton Landmarks in Mathematics

Dantzig, G.B.: Linear programming and extensions, corr ected edn. Princeton Landmarks in Mathematics. Princeton U niv. Press, Princeton (1998)

work page 1998
[8]

Monthly Not

Dehnen, W., Aly, H.: Improving convergence in smoothed p article hydrodynamics simulations without pairing instab ility. Monthly Not. Roy. Astronom. Soc. 425(2), 1068–1082 (2012)

work page 2012
[9]

Fulk, D.A., Quinn, D.W.: An analysis of 1-d smoothed part icle hydrodynamics kernels. J. Comput. Phys. 126(1), 165–180 (1996)

work page 1996
[10]

Monthly Not

Gingold, R.A., Monaghan, J.J.: Smoothed particle hydr odynamics-theory and application to non-spherical stars. Monthly Not. Roy. Astronom. Soc. 181, 375–389 (1977)

work page 1977
[11]

JSIAM Letters 8, 29–32 (2016)

Imoto, Y., Tagami, D.: A truncation error estimate of th e interpolants of a particle method based on the Voronoi deco m- position. JSIAM Letters 8, 29–32 (2016)

work page 2016
[12]

JSIAM Letters 9, 69–72 (2017)

Imoto, Y., Tagami, D.: Truncation error estimates of ap proximate diﬀerential operators of a particle method based on the voronoi decomposition. JSIAM Letters 9, 69–72 (2017)

work page 2017
[13]

Ishijima, K., Kimura, M.: Truncation error analysis of ﬁnite diﬀerence formulae in meshfree particle methods ( in Japanese ). Trans. Japan Soc. Ind. Appl. Math. 20, 165–182 (2010)

work page 2010
[14]

Khayyer, A., Gotoh, H.: Enhancement of stability and ac curacy of the moving particle semi-implicit method. J. Comp ut. Phys. 230(8), 3093–3118 (2011)

work page 2011
[15]

Koshizuka, S.: A particle method for incompressible vi scous ﬂow with ﬂuid fragmentation. J. Comput. Fluid Dyn. 4, 29–46 (1995)

work page 1995
[16]

Nuclear Sci

Koshizuka, S., Oka, Y.: Moving-particle semi-implici t method for fragmentation of incompressible ﬂuid. Nuclear Sci. Eng. 123(3), 421–434 (1996)

work page 1996
[17]

Levin, D.: The approximation power of moving least-squ ares. Math. Comp. 67(224), 1517–1531 (1998)

work page 1998
[18]

Liu, M., Liu, G.: Smoothed particle hydrodynamics (SPH ): an overview and recent developments. Arch. Comput. Metho ds Eng. 17(1), 25–76 (2010) 24 Y. IMOTO

work page 2010
[19]

Astronom

Lucy, L.B.: A numerical approach to the testing of the ﬁs sion hypothesis. Astronom. J. 82, 1013–1024 (1977)

work page 1977
[20]

Mas-Gallic, S., Raviart, P.: A particle method for ﬁrst -order symmetric systems. Numer. Math. 51(3), 323–352 (1987)

work page 1987
[21]

Monaghan, J.J.: Simulating free surface ﬂows with SPH. J. Comput. Phys. 110(2), 399–406 (1994)

work page 1994
[22]

Astr ophys

Monaghan, J.J., Lattanzio, J.C.: A simulation of the co llapse and fragmentation of cooling molecular clouds. Astr ophys. J. 375, 177–189 (1991)

work page 1991
[23]

Murotani, K., Koshizuka, S., Tamai, T., Shibata, K., Mi tsume, N., Yoshimura, S., Tanaka, S., Hasegawa, K., Nagai, E., Fujisawa, T.: Development of hierarchical domain decom position explicit MPS method and application to large-scal e tsunami analysis with ﬂoating objects. J. Adv. Simulation S ci. Eng. 1(1), 16–35 (2014)

work page 2014
[24]

Price, D.J.: Smoothed particle hydrodynamics and magn etohydrodynamics. J. Comput. Phys. 231(3), 759–794 (2012)

work page 2012
[25]

Quinlan, N.J., Basa, M., Lastiwka, M.: Truncation erro r in mesh-free particle methods. Int. J. Numer. Methods Eng. 66(13), 2064–2085 (2006)

work page 2064
[26]

Numeri cal methods in ﬂuid dynamics (Como, 1983), Lect

Raviart, P.A.: An analysis of particle methods. Numeri cal methods in ﬂuid dynamics (Como, 1983), Lect. Notes Math. , vol. 1127, pp. 243–324. Springer, Berlin (1985)

work page 1983
[27]

Schaback, R.: Error estimates and condition numbers fo r radial basis function interpolation. Adv. Comput. Math. 3(3), 251–264 (1995)

work page 1995
[28]

Shakibaeinia, A., Jin, Y.C.: A weakly compressible MPS method for modeling of open-boundary free-surface ﬂow. Int . J. Numer. Meth. Fl. 63(10), 1208–1232 (2010)

work page 2010
[29]

Shakibaeinia, A., Jin, Y.C.: MPS mesh-free particle me thod for multiphase ﬂows. Comput. Methods Appl. Mech. Eng. 229, 13–26 (2012)

work page 2012
[30]

W endland, H.: Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathema tics, vol. 17. Cambridge Univ. Press, Cambridge (2005) 1Institute for Advanced Study, Kyoto UniversityYoshida Konoe- cho, Sakyo-ku, Kyoto 6068501, Japan E-mail address : imoto.yusuke.4e@kyoto-u.ac.jp

work page 2005

[1] [1]

Amicarelli, A., Marongiu, J.C., Leboeuf, F., Leduc, J., Caro, J.: SPH truncation error in estimating a 3D function. C omput. & Fluids 44(1), 279–296 (2011)

work page 2011

[2] [2]

Amicarelli, A., Marongiu, J.C., Leboeuf, F., Leduc, J., Neuhauser, M., Fang, L., Caro, J.: SPH truncation error in estimating a 3D derivative. Int. J. Numer. Methods Eng. 87(7), 677–700 (2011)

work page 2011

[3] [3]

Methods Appl

Ben Moussa, B.: On the convergence of SPH method for scala r conservation laws with boundary conditions. Methods Appl . Anal. 13(1), 29–62 (2006)

work page 2006

[4] [4]

Ben Moussa, B., Vila, J.: Convergence of SPH method for sc alar nonlinear conservation laws. SIAM J. Numer. Anal. 37(3), 863–887 (2000)

work page 2000

[5] [5]

Benz, W., Asphaug, E.: Simulations of brittle solids usi ng smooth particle hydrodynamics. Comput. Phys. Commun. 87(1), 253–265 (1995)

work page 1995

[6] [6]

Ca mbridge Univ

Boissonnat, J.D., Yvinec, M.: Algorithmic geometry. Ca mbridge Univ. Press, Cambridge (1998)

work page 1998

[7] [7]

Princeton Landmarks in Mathematics

Dantzig, G.B.: Linear programming and extensions, corr ected edn. Princeton Landmarks in Mathematics. Princeton U niv. Press, Princeton (1998)

work page 1998

[8] [8]

Monthly Not

Dehnen, W., Aly, H.: Improving convergence in smoothed p article hydrodynamics simulations without pairing instab ility. Monthly Not. Roy. Astronom. Soc. 425(2), 1068–1082 (2012)

work page 2012

[9] [9]

Fulk, D.A., Quinn, D.W.: An analysis of 1-d smoothed part icle hydrodynamics kernels. J. Comput. Phys. 126(1), 165–180 (1996)

work page 1996

[10] [10]

Monthly Not

Gingold, R.A., Monaghan, J.J.: Smoothed particle hydr odynamics-theory and application to non-spherical stars. Monthly Not. Roy. Astronom. Soc. 181, 375–389 (1977)

work page 1977

[11] [11]

JSIAM Letters 8, 29–32 (2016)

Imoto, Y., Tagami, D.: A truncation error estimate of th e interpolants of a particle method based on the Voronoi deco m- position. JSIAM Letters 8, 29–32 (2016)

work page 2016

[12] [12]

JSIAM Letters 9, 69–72 (2017)

Imoto, Y., Tagami, D.: Truncation error estimates of ap proximate diﬀerential operators of a particle method based on the voronoi decomposition. JSIAM Letters 9, 69–72 (2017)

work page 2017

[13] [13]

Ishijima, K., Kimura, M.: Truncation error analysis of ﬁnite diﬀerence formulae in meshfree particle methods ( in Japanese ). Trans. Japan Soc. Ind. Appl. Math. 20, 165–182 (2010)

work page 2010

[14] [14]

Khayyer, A., Gotoh, H.: Enhancement of stability and ac curacy of the moving particle semi-implicit method. J. Comp ut. Phys. 230(8), 3093–3118 (2011)

work page 2011

[15] [15]

Koshizuka, S.: A particle method for incompressible vi scous ﬂow with ﬂuid fragmentation. J. Comput. Fluid Dyn. 4, 29–46 (1995)

work page 1995

[16] [16]

Nuclear Sci

Koshizuka, S., Oka, Y.: Moving-particle semi-implici t method for fragmentation of incompressible ﬂuid. Nuclear Sci. Eng. 123(3), 421–434 (1996)

work page 1996

[17] [17]

Levin, D.: The approximation power of moving least-squ ares. Math. Comp. 67(224), 1517–1531 (1998)

work page 1998

[18] [18]

Liu, M., Liu, G.: Smoothed particle hydrodynamics (SPH ): an overview and recent developments. Arch. Comput. Metho ds Eng. 17(1), 25–76 (2010) 24 Y. IMOTO

work page 2010

[19] [19]

Astronom

Lucy, L.B.: A numerical approach to the testing of the ﬁs sion hypothesis. Astronom. J. 82, 1013–1024 (1977)

work page 1977

[20] [20]

Mas-Gallic, S., Raviart, P.: A particle method for ﬁrst -order symmetric systems. Numer. Math. 51(3), 323–352 (1987)

work page 1987

[21] [21]

Monaghan, J.J.: Simulating free surface ﬂows with SPH. J. Comput. Phys. 110(2), 399–406 (1994)

work page 1994

[22] [22]

Astr ophys

Monaghan, J.J., Lattanzio, J.C.: A simulation of the co llapse and fragmentation of cooling molecular clouds. Astr ophys. J. 375, 177–189 (1991)

work page 1991

[23] [23]

Murotani, K., Koshizuka, S., Tamai, T., Shibata, K., Mi tsume, N., Yoshimura, S., Tanaka, S., Hasegawa, K., Nagai, E., Fujisawa, T.: Development of hierarchical domain decom position explicit MPS method and application to large-scal e tsunami analysis with ﬂoating objects. J. Adv. Simulation S ci. Eng. 1(1), 16–35 (2014)

work page 2014

[24] [24]

Price, D.J.: Smoothed particle hydrodynamics and magn etohydrodynamics. J. Comput. Phys. 231(3), 759–794 (2012)

work page 2012

[25] [25]

Quinlan, N.J., Basa, M., Lastiwka, M.: Truncation erro r in mesh-free particle methods. Int. J. Numer. Methods Eng. 66(13), 2064–2085 (2006)

work page 2064

[26] [26]

Numeri cal methods in ﬂuid dynamics (Como, 1983), Lect

Raviart, P.A.: An analysis of particle methods. Numeri cal methods in ﬂuid dynamics (Como, 1983), Lect. Notes Math. , vol. 1127, pp. 243–324. Springer, Berlin (1985)

work page 1983

[27] [27]

Schaback, R.: Error estimates and condition numbers fo r radial basis function interpolation. Adv. Comput. Math. 3(3), 251–264 (1995)

work page 1995

[28] [28]

Shakibaeinia, A., Jin, Y.C.: A weakly compressible MPS method for modeling of open-boundary free-surface ﬂow. Int . J. Numer. Meth. Fl. 63(10), 1208–1232 (2010)

work page 2010

[29] [29]

Shakibaeinia, A., Jin, Y.C.: MPS mesh-free particle me thod for multiphase ﬂows. Comput. Methods Appl. Mech. Eng. 229, 13–26 (2012)

work page 2012

[30] [30]

W endland, H.: Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathema tics, vol. 17. Cambridge Univ. Press, Cambridge (2005) 1Institute for Advanced Study, Kyoto UniversityYoshida Konoe- cho, Sakyo-ku, Kyoto 6068501, Japan E-mail address : imoto.yusuke.4e@kyoto-u.ac.jp

work page 2005