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
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DOI：
10.1016/j.cam.2015.03.012
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发表时间：
2015-10
期刊：
影响因子：
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通讯作者：
M. Dehghan;Mostafa Abbaszadeh;A. Mohebbi
中科院分区：
文献类型：
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作者：
M. Dehghan;Mostafa Abbaszadeh;A. Mohebbi
本文提出了一种求解非线性广义Benjamin-Bona-Mahony-Burgers方程和正则长波方程的数值方法。首先，通过有限差分公式逼近时间导数得到时间离散格式，然后利用无插值单元伽辽金方法逼近空间导数。无单元伽辽金方法使用一种弱形式的考虑方程，它类似于有限元方法，不同之处在于，在无单元伽辽金方法中，试验函数和试验函数是移动最小二乘近似形状函数。此外，在无单元Galerkin方法中，我们不使用任何三角形，四边形或其他类型的网格。它是一种全局方法，而有限元法是一种局部方法。无单元伽辽金方法并不是一种真正的无网格方法，它采用背景网格进行积分。利用能量法证明了时间离散格式在时间变量上是无条件稳定和收敛的。我们证明了时间离散格式的收敛阶为O （τ）。由于移动最小二乘逼近的形状函数不具有Kronecker特性，我们不能直接实现基本的边界条件。因此，采用具有上述性质的改进的移动最小二乘形状函数。对本文提出的方法进行了误差估计。同时，对两个方程在不同复杂几何上的二维形式进行了求解。本文的目的是证明基于弱形式的无网格方法也适用于处理非线性偏微分方程，并得到新方法的误差界。数值算例验证了该方法的有效性。
In 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 Galerkin method, we do not use any triangular, quadrangular or other type of meshes. It is a global method while finite element 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 do not have Kronecker delta property, we cannot 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.