论文标题
用于扩散和细胞扩散方程的高阶BDF方法的平行算法
A parallel-in-time algorithm for high-order BDF methods for diffusion and subdiffusion equations
论文作者
论文摘要
在本文中，我们提出了一种平行算法，用于求解抛物线方程。特别是，我们应用了$ k $ step的向后分化公式，然后使用波形放松技术来开发迭代求解器。每种结果迭代都代表一个周期性的系统，可以使用对角技术并行进一步求解。通过使用生成函数方法，在理论上检查了波形弛豫迭代的收敛性。我们在本文中建立的方法扩展了Gander和Wu [Numer中的单步方法的现有参数。 Math。，143（2019），第489--527页，至订单六的一般BDF方法。该参数可以进一步应用于时间分数子扩散方程，因为由于分数差分运算符的非局部性，其离散化具有标准BDF方法的共同属性。提出了说明性的数值结果以补充理论分析。
In this paper, we propose a parallel-in-time algorithm for approximately solving parabolic equations. In particular, we apply the $k$-step backward differentiation formula, and then develop an iterative solver by using the waveform relaxation technique. Each resulting iteration represents a periodic-like system, which could be further solved in parallel by using the diagonalization technique. The convergence of the waveform relaxation iteration is theoretically examined by using the generating function method. The approach we established in this paper extends the existing argument of single-step methods in Gander and Wu [Numer. Math., 143 (2019), pp. 489--527] to general BDF methods up to order six. The argument could be further applied to the time-fractional subdiffusion equation, whose discretization shares common properties of the standard BDF methods, because of the nonlocality of the fractional differential operator. Illustrative numerical results are presented to complement the theoretical analysis.
