The improved complex variable element-free Galerkin method for two-dimensional Schrödinger equation

Zhang, Luwen; Deng, Yajie; Liew, K.M.; Cheng, Yumin

doi:10.1016/j.camwa.2014.07.024

Cited by 44 publications

(19 citation statements)

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“…The main aim of [72] is to numerical study the degenerate parabolic equations arising from the spatial diffusion of biological populations based on the improved EFG method. Authors of [73] proposed a numerical algorithm for solving two-dimensional Schrödinger equation using the ICVEFG method. An improved complex variable MLS Ritz (ICVMLS-Ritz) method is proposed in [74] for obtaining numerical solution of the 2D nonlinear Schrödinger equation.…”

Section: A Brief Review For Element-free Galerkin (Efg)mentioning

confidence: 99%

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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

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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 For Element-free Galerkin (Efg)mentioning

confidence: 99%

“…The main aim of [67] is to illustrate a numerical algorithm based on the improved complex variable EFG (ICVEFG) method for solving two-dimensional Schrödinger equation. The ICVEFG method employs the improved complex variable MLS (ICVMLS) as test and trial functions in the obtained variational form.…”

Section: A Brief Review For Element-free Galerkin (Efg)mentioning

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

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“…A numerical study of two-dimensional Schrödinger equation is carried out in [69] using the improved complex variable EFG (ICVEFG) method. The ICVEFG method involves employment of the improved complex variable MLS (ICVMLS) in the EFG procedure.…”

Section: A Brief Review For Element Free Galerkin (Efg)mentioning

confidence: 99%

Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach

Dehghan

Abbaszadeh

2016

Computers & Mathematics with Applications

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Available online xxxx Keywords: Element free Galerkin (EFG) and movingKriging interpolation Fourth-order Runge-Kutta method Mathematical biology and Turing system Gray-Scott model and Allen-Cahn equation Ginzburg-Landau model and Brusselator equation Predator-prey model with additional food supply to predator a b s t r a c tIn this paper a numerical procedure is presented for solving a class of three-dimensional Turing system. First, we discrete the spatial direction using element free Galerkin (EFG) method based on the shape functions of moving Kriging interpolation. Then, to achieve a high-order accuracy, we use the fourth-order exponential time differencing Runge-Kutta (ETDRK4) method. Using this discretization for the temporal dimension, we obtain an explicit scheme and do not need to solve nonlinear system of equations. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Also building shape functions of MLS approximation is a time consuming procedure. Because of the mentioned reasons we employ the shape functions of moving Kriging interpolation technique which have the mentioned property and less CPU time is required for building them. For testing this method on threedimensional PDEs, we select some equations and system of PDEs such as Allen-Cahn, Gray-Scott, Ginzburg-Landau, Brusselator models, predator-prey model with additional food supply to predator. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme.

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“…Unlike the FE method, the element-free method furnishes solutions in terms nodes instead of meshes [35]. Recently, a number of element-free methods have been proposed as numerical techniques based on different sets of trail functions such as the element-free Galerkin method [36], smooth particle hydrodynamics method [37], radial basis function method [38][39][40][41], meshless local Petrov-Galerkin method [42], reproducing kernel particle method [43] and improved element-free Galerkin (IEFG) method [44,45]. In particular, the IEFG method employs improved moving least-squares (IMLS) to overcome the drawback of the moving least-squares (MLS) approximation [46][47][48][49].…”

Section: Introductionmentioning

confidence: 99%

Geometrically nonlinear large deformation analysis of functionally graded carbon nanotube reinforced composite straight-sided quadrilateral plates

Zhang

Liew

2015

Computer Methods in Applied Mechanics and Engineering

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The improved complex variable element-free Galerkin method for two-dimensional Schrödinger equation

Cited by 44 publications

References 41 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

Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach

Geometrically nonlinear large deformation analysis of functionally graded carbon nanotube reinforced composite straight-sided quadrilateral plates

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