Numerical solutions of two-dimensional flow fields by using the localized method of approximate particular solutions

Fan, Chia-Ming; Yang, Chi-Hung; Lai, Wei-Shiang

doi:10.1016/j.enganabound.2015.03.012

Cited by 13 publications

(3 citation statements)

References 30 publications

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“…where c 1 and c 2 are coefficients to be determined. By substituting (18) into (17), and comparing the coefficients on the both sides, we have…”

Section: Basis Function: Single Trigonometric Functionmentioning

confidence: 99%

“…Besides, the localized MPS (LMPS) has been recently proposed by Yao et al [17] for solving large-scale problems. The LMPS has been successfully applied to velocity-vorticity formulation of the Navier-Stokes equations by Fan et al [18]. From the above discussions, it could be realized that the newly-developed MPS is getting matured and has great potential to be extended to solving real-life problems.…”

Section: Introductionmentioning

confidence: 99%

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The method of particular solutions using trigonometric basis functions

Tian

Fan

et al. 2018

Journal of Computational and Applied Mathematics

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In this paper, the method of particular solutions (MPS) using trigonometric functions as the basis functions is proposed to solve two-dimensional elliptic partial differential equations. The inhomogeneous term of the governing equation is approximated by Fourier series and the closed-form particular solutions of trigonometric functions are derived using the method of undetermined coefficients. Once the particular solutions for the trigonometric basis functions are derived, the standard MPS can be applied for solving partial differential equations. In comparing with the use of radial basis functions and polynomials in the MPS, our proposed approach provide another simple approach to effectively solving two-dimensional elliptic partial differential equations. Five numerical examples are provided in this paper to validate the merits of the proposed meshless method.

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“…where c 1 and c 2 are coefficients to be determined. By substituting (18) into (17), and comparing the coefficients on the both sides, we have…”

Section: Basis Function: Single Trigonometric Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The method of particular solutions using trigonometric basis functions

Tian

Fan

et al. 2018

Journal of Computational and Applied Mathematics

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“…In the LMAPS, the k-d tree algorithm [38] is applied for searching the neighboring points of each local center. Recently, the LMAPS has been successfully applied for the simulation of heat conduction, the molecular dynamics, the wave propagation, and the flow fields [39][40][41][42].…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of Non-Linear Schrödinger Equations in Arbitrary Domain by the Localized Method of Approximate Particular Solution

Hong¹

2019

AAMM

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The aim of this paper is to propose a fast meshless numerical scheme for the simulation of non-linear Schrödinger equations. In the proposed scheme, the implicit-Euler scheme is used for the temporal discretization and the localized method of approximate particular solution (LMAPS) is utilized for the spatial discretization. The multiple-scale technique is introduced to obtain the shape parameters of the multiquadric radial basis function for 2D problems and the Gaussian radial basis function for 3D problems. Six numerical examples are carried out to verify the accuracy and efficiency of the proposed scheme. Compared with well-known techniques, numerical results illustrate that the proposed scheme is of merits being easy-to-program, high accuracy, and rapid convergence even for long-term problems. These results also indicate that the proposed scheme has great potential in large scale problems and real-world applications.

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