Numerical Solution of the Nonlocal Diffusion Equation on the Real Line
Numerical Solution of the Nonlocal Diffusion Equation on the Real Line
复制标题
实线上非局域扩散方程的数值解
DOI：
10.1137/16m1090107
复制
发表时间：
2017-09
影响因子：
3.1
通讯作者：
Zhang Jiwei
中科院分区：
文献类型：
--
作者：
Zheng Chunxiong;Hu Jiashun;Du Qiang;Zhang Jiwei
本文考虑实轴上非局部扩散方程的数值计算。我们首先应用一个被广泛研究的求积格式在无界区域上得到一个离散的非局部扩散系统。然后，我们基于$z$变换的谱分析，推导出离散问题的另一种形式。这种新的表述可以看作是一个定义在有界域上的带有人工边界条件的系统，它允许我们将原来的无限域问题转化为等价的有界域问题。为了数值实现精确的人工边界条件，我们应用梯形求积规则来逼近由逆$z$变换引起的轮廓积分。数值算例验证了该方法的有效性。
Numerical computation of a nonlocal diffusion equation on the real axis is considered in this paper. We first apply an extensively studied quadrature scheme to obtain a discrete nonlocal diffusion system on an unbounded domain. Then we derive an alternative formulation of the discrete problem based on the spectral analysis of the $z$-transform. This new formulation can be seen as a system defined on a bounded domain with an artificial boundary condition, and it allows us to reformulate the original infinite domain problem into an equivalent bounded domain problem. To numerically implement the exact artificial boundary condition, we apply the trapezoidal quadrature rule to approximate the contour integral induced by the inverse $z$-transform. Numerical examples are presented to demonstrate the effectiveness of our approach.