Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations

Thounthong, Phatiphat; Khan, Muhammad Nawaz; Hussain, Ibrar; Ahmad, Imtiaz

doi:10.3390/math6120327

Cited by 30 publications

(15 citation statements)

References 21 publications

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“…The numerical results of RBF-based algorithms have demonstrated that they are truly meshless, accurate, and easy to implement. Some interesting models can be found in [26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

et al. 2019

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In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial discretization, which convert the mentioned partial differential equations into a system of ordinary differential equations. The obtained system has been solved by the forward Euler difference formula. An upwind technique is utilized in the case of the convection-dominated coupled Burgers’ model equation. Having no need for the mesh in the problem domain and being less sensitive to the variation of the shape parameter as compared to global meshless methods are the salient features of the local meshless method. Both rectangular and non-rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy, and the ease of implementation of the proposed method are shown via test problems.

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Section: Introductionmentioning

confidence: 99%

Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

et al. 2019

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“…The numerical results of RBFs based algorithms have demonstrated that it is truly meshfree, accurate and east to implement. Some interesting models can be found in [30,[33][34][35][36][37][38][39][40].…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of Partial Differential Equations via Local Meshless Method

et al. 2019

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In this paper, numerical simulation of one, two and three dimensional partial differential equations (PDEs) are obtained by local meshless method using radial basis functions (RBFs). Both local and global meshless collocation procedures are used for spatial discretization, which convert the given PDEs into a system of ODEs. Multiquadric, Gaussian and inverse quadratic RBFs are used for spatial discretization. The obtained system of ODEs has been solved by different time integrators. The salient feature of the local meshless method (LMM) is that it does not require mesh in the problem domain and also far less sensitive to the variation of shape parameter as compared to the global meshless method (GMM). Both rectangular and non rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy and ease implementation of the proposed method are shown via test problems.

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“…The horizon of meshless methods is continuously expanding and their applicability is increasing rapidly due to their ease of implementation in higher dimensions on a set of uniform or scattered data points in regular and irregular geometries. In the last few years, it is observed that meshless methods have been extensively employed for numerical simulations of different types of PDEs [29,[42][43][44]. Meshless methods reduce complexity caused due to dimensionality to a large extent which is being faced in the carrying out of conventional methods like finite-element and finite-difference procedures.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Local Formulation for Time–Dependent PDEs

et al. 2019

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In this paper, a local meshless method (LMM) based on radial basis functions (RBFs) is utilized for the numerical solution of various types of PDEs. This local approach has flexibility with respect to geometry along with high order of convergence rate. In case of global meshless methods, the two major deficiencies are the computational cost and the optimum value of shape parameter. Therefore, research is currently focused towards localized RBFs approximations, as proposed here. The proposed local meshless procedure is used for spatial discretization, whereas for temporal discretization, different time integrators are employed. The proposed local meshless method is testified in terms of efficiency, accuracy and ease of implementation on regular and irregular domains.

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Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations

Cited by 30 publications

References 21 publications

Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

Numerical Simulation of Partial Differential Equations via Local Meshless Method

An Efficient Local Formulation for Time–Dependent PDEs

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