A fast and efficient two-grid method for solving d-dimensional poisson equations
A fast and efficient two-grid method for solving d-dimensional poisson equations
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DOI：
10.1007/s11075-015-0057-8
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发表时间：
2016-07
影响因子：
2.1
通讯作者：
Hamid Moghaderi;M. Dehghan;M. Hajarian
中科院分区：
文献类型：
--
作者：
Hamid Moghaderi;M. Dehghan;M. Hajarian
本文的目的是介绍一种新的快速有效的双网格方法来求解d维（d= 1，2，3）Poisson椭圆型方程。所有内部网格点上的有限差分方程构成一个大型稀疏线性系统，需要高效求解。这种稀疏线性系统的解决方案的成本通常占主导地位的离散偏微分方程求解的总成本。有限差分方程的基础上应用有限差分计划的二阶和四阶（紧凑有限差分法）离散的空间导数。用一种新的两重网格方法（NTG）求解了所得到的Poisson椭圆方程组，并指出NTG方法在求解大型稀疏线性方程组时比标准两重网格方法更快、更有效。我们利用局部傅立叶分析表明，新的两个网格方法的光谱半径的一维和二维模型是小于标准的两个网格方法。并将相应的算法扩展到新的多重网格法。数值算例表明了新算法求解d维Poisson方程的有效性。
The aim of this paper is to introduce a fast and efficient new two-grid method to solve thed-dimensional (d=1,2,3) Poisson elliptic equations. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The finite difference equations are based on applying a finite difference scheme of two- and four-orders (compact finite difference method) for discretizing the spatial derivative. The obtained linear systems of Poisson elliptic equations have been solved by a new two-grid (NTG) method and we also note that the NTG method which is used for solving the large sparse linear systems is faster and more effective than that of the standard two-grid method. We utilize the local Fourier analysis to show that the spectral radius of the new two-grid method for 1D and 2D models is less than that of the standard two-grid method. As well as, we expand the corresponding algorithm to the new multi-grid method. The numerical examples show the efficiency of the new algorithms for solving thed-dimensional Poisson equations.