扩散和次扩散方程高阶BDF方法的时间并行算法
A parallel-in-time algorithm for high-order BDF methods for diffusion and subdiffusion equations
摘要：
本文提出了一种近似求解抛物型方程的时间并行算法。特别是，我们应用$k$-步向后微分公式，然后开发一个迭代求解器，通过使用波形松弛技术。每一次迭代都代表一个类牛顿系统，可以通过对角化技术进一步并行求解。本文用母函数法从理论上证明了波形松弛迭代法的收敛性。本文建立的方法扩展了Gander和Wu [Numer.]中单步方法的现有论点。数学、143（2019），pp. 489--527]到六阶的一般BDF方法。由于分数阶微分算子的非局部性，该方法可以进一步应用于时间分数阶次扩散方程，其离散化具有标准BDF方法的共同性质.说明性的数值结果，以补充理论分析。
英文摘要：
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.
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