Influence of several factors in the generalized finite difference method

Benito, Juan José; Ureña, Francisco; Gavete, L.

doi:10.1016/s0307-904x(01)00029-4

Cited by 262 publications

(138 citation statements)

References 9 publications

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“…An evolution of the method of finite differences has been the development of generalized finite difference method (GFDM) that can be applied as a meshless or meshfreee method to irregular grids or clouds of points. Benito, Ureña and Gavete have made interesting contributions to the development of this method [2][3][4][5][6][7]. This paper shows the application of the generalized finite difference method to solve the advection-diffusion equation by an explicit method.…”

Section: Introductionmentioning

confidence: 94%

“…If * () Buis minimized with respect to the partial derivatives up to second order, the following linear equation system is defined 2 2 "Boundary conditions are neglected by the von Neumann method which applies in theory only to pure initial value problems with periodic initial data. It does however provide necessary conditions for stability of constant coefficient problems regardless of the type of boundary condition" [9].…”

Section: Convergencementioning

confidence: 99%

See 1 more Smart Citation

Modelling of the advection–diffusion equation with a meshless method without numerical diffusion

Gavete

Ureña

Benito

et al. 2012

International Journal of Computer Mathematics

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A comprehensive study is presented regarding the stability of the forward explicit integration technique with generalized finite difference spatial discretizations, free of numerical diffusion, applied to the advection-diffusion equation. The modified equivalent partial differential equation approach is used to demonstrate that the approximation is free of numerical diffusion. Two-dimensional results are obtained using the von Neumann method of stability analysis. Numerical results are presented showing the accuracy obtained.

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Section: Introductionmentioning

confidence: 94%

Section: Convergencementioning

confidence: 99%

Modelling of the advection–diffusion equation with a meshless method without numerical diffusion

Gavete

Ureña

Benito

et al. 2012

International Journal of Computer Mathematics

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“…Recently, Benito, Urena and Gavete gave a discussion about the influence of several factors in the GFD method, and a comparison between the GFD method and the EFGM in solving Laplace equation is also presented in [49][50][51]. The GFD method shows more accurate than the EFGM.…”

Section: Introductionmentioning

confidence: 99%

A Lagrangian Approach for Computational Acoustics with Meshfree Method

Zhang

Smith

Zhang

et al. 2017

Preprint

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Abstract:Although Eulerian approaches are standard in computational acoustics, they are less effective for certain classes of problems like bubble acoustics and combustion noise. A different approach for solving acoustic problems is to compute with individual particles following particle motion. In this paper, a Lagrangian approach to model sound propagation in moving fluid is presented and implemented numerically, using three meshfree methods to solve the Lagrangian acoustic perturbation equations (LAPE) in the time domain. The LAPE split the fluid dynamic equations into a set of hydrodynamic equations for the motion of fluid particles and perturbation equations for the acoustic quantities corresponding to each fluid particle. Then, three meshfree methods, the smoothed particle hydrodynamics (SPH) method, the corrective smoothed particle (CSP) method, and the generalized finite difference (GFD) method, are introduced to solve the LAPE and the linearized LAPE (LLAPE). The SPH and CSP methods are widely used meshfree methods, while the GFD method based on the Taylor series expansion can be easily extended to higher orders. Applications to modeling sound propagation in steady or unsteady fluids in motion are outlined, treating a number of different cases in one and two space dimensions. A comparison of the LAPE and the LLAPE using the three meshfree methods is also presented. The Lagrangian approach shows good agreement with exact solutions. The comparison indicates that the CSP and GFD method exhibit convergence in cases with different background flow. The GFD method is more accurate, while the CSP method can handle higher Courant numbers.

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“…The overview of the FVM applications in rock mechanics can be seen in [5]. The latest improvements of the FDM/FVM can be seen in [6][7][8][9]. One of the most well-known computer programs for solving a variety of rock mechanics problems by FDM/FVM is FLAC/FLAC3D [10].…”

Section: Introductionmentioning

confidence: 99%

Overview of the numerical methods for the modelling of rock mechanics problems

2016

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Subject reviewThe numerical methods have their origin in the early 1960s and even at that time it was noted that numerical methods can be successfully applied in various engineering and scientific fields, including the rock mechanics. Moreover, the rapid development of computers was a necessary background for solving computationally more demanding problems and the development process of the methods in general. Thus, we have many different methods presently, which can be separated into two main branches: continuum and discontinuum-based numerical methods. Some problems require the strengths of both main approaches which brought the hybrid continuum/discontinuum methods. The first goal of this paper is to present the state of the art numerical methods and approaches for solving the rock mechanics problems, as well as to give the brief explanation about the theoretical background of each method. The second goal is to emphasise the area of applicability of the methods in rock mechanics. Keywords: numerical methods; numerical modelling; rock mechanics Pregled numeričkih metoda za modeliranje u mehanici stijenaPregledni rad Počeci numeričkih metoda sežu u rane 1960-e. Već tada je bilo jasno da numeričke metode mogu biti uspješno upotrijebljene za različita inženjerska i znanstvena područja, uključujući i primjenu u mehanici stijena. Ubrzan razvoj računala je omogućavao razvoj numeričkih metoda i rješavanje računalno zahtjevnijih sustava. Takav razvoj doveo je do velikog broja različitih metoda i pristupa koji se mogu svrstati u dvije skupine: metode kontinuuma i metode diskontinuuma. Određene zadaće zahtijevaju prednosti oba pristupa, što je dovelo do razvoja kombiniranih konačno-diskretnih metoda. Prvi cilj ovog rada je predstavljanje numeričkih metoda i pristupa koji se koriste za rješavanje zadaća u mehanici stijena, kao i kratko objašnjenje osnovnih teorijskih postavki svake od metoda. Drugi cilj je osvrt na primjenjivost pojedine metode u mehanici stijena.

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Influence of several factors in the generalized finite difference method

Cited by 262 publications

References 9 publications

Modelling of the advection–diffusion equation with a meshless method without numerical diffusion

Modelling of the advection–diffusion equation with a meshless method without numerical diffusion

A Lagrangian Approach for Computational Acoustics with Meshfree Method

Overview of the numerical methods for the modelling of rock mechanics problems

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