A Meshless and Parallelizable Method for Differential Equations with Time-Delay
A Meshless and Parallelizable Method for Differential Equations with Time-Delay
复制标题
时滞微分方程的无网格并行化方法
DOI：
10.4208/nmtma.2018.m1636
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Huang Chengming
中科院分区：
文献类型：
--
作者：
Wu Shulin;Huang Chengming
数值计算在时滞微分方程的研究中起着重要的作用，因为通常无法得到简单而显式的解析解，基于离散时间导数的时间步进方法是解决这一问题的主要工具。为了得到精确的数值解，有必要要求：这将是一个相当不受欢迎的限制，什么时候？延时量小。在这次演讲中，我们提出了一种方法来解决一类时滞问题，这是完全无网格的。其思想是在复平面上沿沿着一条精心设计的轮廓线，用拉普拉斯逆变换来表示解，然后用Filon-Clenshaw-Curits（FCC）求积法在几个快速增长的子区间上逼近轮廓积分。所有感兴趣的时间点的解的计算自然是可并行的，并且对于每个时间点，所有子区间中的FCC求积的实现也是可并行的。对于每个时间点和每个子区间，可以通过快速傅里叶变换来实现FCC正交。我们提出的方法和稳定性分析的FCC正交的权重计算在我们的情况下，所提出的误差界。数值结果验证了该方法的有效性。
Numerical computation plays an important role in the study of differential equations with time-delay, because a simple and explicit analytic solution is usually unavailable. Time-stepping methods based on discretizing the temporal derivative with some step-size ∆ t are the main tools for this task. To get accurate numerical solutions, it is necessary to require ∆ t < ? t and this will be a rather unwelcome restriction when ? , quantity of time-delay, is small. In this talk, we propose a method for a class of time-delay problems, which is completely meshless. The idea lies in representing the solution by its Laplace inverse transform along a carefully designed contour in the complex plane and then approximating the contour integral by the Filon-Clenshaw-Curits (FCC) quadrature in a few fast growing subintervals. The computations of the solution for all time points of interest are naturally parallelizable and for each time point the implementations of the FCC quadrature in all subintervals are also parallelizable. For each time point and each subinterval, the FCC quadrature can be implemented by fast Fourier transform. We present error bounds for the proposed method and stability analysis for the computation of the weights of the FCC quadrature in our situation. Numerical results are given to validate the efficiency of the proposed method.