A local radial basis function method for advection–diffusion–reaction equations on complexly shaped domains

Sarra, Scott A.

doi:10.1016/j.amc.2012.03.062

Cited by 120 publications

(68 citation statements)

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“…The method of local radial basis functions is employed in [47] for obtaining the numerical solution of the advection-diffusion-reaction equation. Sarra in [48] used a regularization method to prevent the failure of the Cholesky factorization and improved the accuracy of symmetric positive definite (SPD) matrix factorizations when the matrices are severely ill-conditioned.…”

Section: A Brief Review Of the Meshless Methodsmentioning

confidence: 99%

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

Dehghan

Abbaszadeh

Mohebbi

2015

Journal of Computational and Applied Mathematics

103

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Please cite this article as: M. Dehghan, M. Abbaszadeh, A. Mohebbi, The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate, Journal of Computational and Applied Mathematics (2015), http://dx. AbstractIn this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the interpolating element-free Galerkin approach to approximate the spatial derivatives. The element-free Galerkin method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the element-free Galerkin method test and trial functions are moving least squares approximation shape functions. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. It is a global method while finite elements method is a local one. The element free Galerkin method is not a truly meshless method and for integration employs a background mesh. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O(τ ). Since the shape functions of moving least squares approximation don't have Kronecker delta property, we can not implement the essential boundary condition, directly. Thus, the improved moving least squares shape functions that have the mentioned property are employed . An error estimate for the method proposed in the current paper is obtained. Also, the two-dimensional version of both equations on different complex geometries is solved. The aim of this paper is to show that the meshless method based on the weak form is also suitable for the treatment of the nonlinear partial differential equations and to obtain an error bound for the new method. Numerical examples confirm the efficiency of the proposed scheme. (M. Dehghan), a mohebbi@kashanu.ac.ir (A. Mohebbi), m.abbaszadeh@aut.ac.ir (M. Abbaszadeh).

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Section: A Brief Review Of the Meshless Methodsmentioning

confidence: 99%

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

Dehghan

Abbaszadeh

Mohebbi

2015

Journal of Computational and Applied Mathematics

103

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“…According to numerical evidence, RBF methods usually provide better accuracy if their system matrix is ill-conditioned. For computation with double precision, the shape parameter can be selected for each stencil so that the condition number is between 10 13 and 10 15 [12] because the error curve of the corresponding range of the shape parameter is the smooth curve before beginning to oscillate after that, as shown in Figure 2a. The following pseudocode will be used to select the shape parameter in this problem so that the condition number is in the desired range: K = 0, Kmin = 1.0 × 10 13 , Kmax = 1.0 × 10 15 , shape = initial shape, increment = 0.01 while K < Kmin or K > Kmax : form B K = condition number of B if K < Kmin shape = shape -increment elseif K > Kmax shape = shape + increment.…”

Section: Methods 3 : Rbf-fd Methods With Preconditioning Strategymentioning

confidence: 99%

“…After the calculation at all interior nodes, the approximate solution can be computed from the linear system of equations, for which the RBF-FD system matrix is sparse and therefore can be effectively inverted. Note that the RBF-FD method that we described can be efficiently and applicably implemented with large scale practical problems such as the global electric circuit within the Earth's atmosphere problem [8], steady problems in solid and fluid mechanics [9][10][11], chemotaxis models [12], and diffusion problems [13,14].…”

Section: Introductionmentioning

confidence: 99%

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

Yensiri

Skulkhu

2017

Mathematics

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Abstract:The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies.

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“…Bayona et al [6,7] proposed two methods to compute the optimal shape parameter of MQ-FD method for solving ordinary and time-independent partial differential equations. In some literature the RBF-FD method is used with other names such as local radial basis function method [35] and local radial basis functions based differential quadrature collocation method [15].…”

Section: Introductionmentioning

confidence: 99%

Stability and convergence of radial basis function finite difference method for the numerical solution of the reaction–diffusion equations

Golbabai

Nikpour

2015

Applied Mathematics and Computation

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A local radial basis function method for advection–diffusion–reaction equations on complexly shaped domains

Cited by 120 publications

References 50 publications

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

Stability and convergence of radial basis function finite difference method for the numerical solution of the reaction–diffusion equations

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