Geometric discretization of the multidimensional Dirac delta distribution - Application to the Poisson equation with singular source terms
Geometric discretization of the multidimensional Dirac delta distribution - Application to the Poisson equation with singular source terms
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DOI：
10.1016/j.jcp.2017.06.003
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发表时间：
2017-10-01
影响因子：
4.1
通讯作者：
Gibou, Frederic
中科院分区：
文献类型：
--
作者：
Egan, Raphael;Gibou, Frederic
提出了一种多维狄拉克分布的离散化方法。我们证明了它在积分问题的背景下的适用性，以及在具有恒定或可变扩散系数的泊松方程中离散狄拉克分布源项的适用性。离散化是基于单元的，因此可以直接应用于四叉树/八叉树网格。该方法可得到二阶精度的积分结果。当用它来模拟泊松方程中的狄拉克分布源项时，观察到超线性收敛：观察到的收敛阶为2或稍小。该方法与文献[1,2]中提出的余维1曲面的Dirac δ分布离散化一致。我们提出了四叉树/八叉树的构造方法以保持收敛性，并给出了各种数值示例，包括难以处理的均匀网格的多尺度问题。(C) 2017爱思唯尔公司版权所有。
We present a discretization method for the multidimensional Dirac distribution. We show its applicability in the context of integration problems, and for discretizing Dirac-distributed source terms in Poisson equations with constant or variable diffusion coefficients. The discretization is cell-based and can thus be applied in a straightforward fashion to Quadtree/Octree grids. The method produces second-order accurate results for integration. Superlinear convergence is observed when it is used to model Dirac-distributed source terms in Poisson equations: the observed order of convergence is 2 or slightly smaller. The method is consistent with the discretization of Dirac delta distribution for codimension one surfaces presented in [1,2]. We present Quadtree/Octree construction procedures to preserve convergence and present various numerical examples, including multi-scale problems that are intractable with uniform grids. (C) 2017 Elsevier Inc. All rights reserved.