大规模延迟微分方程组卷积Schwarz波形松弛算法收敛性研究

Schwarz waveform relaxation is a new parallel numerical method proposed in recent ten years, which got considerable attention. In this project we continue to study the convergence of the Schwarz waveform relaxation algorithm for several representative delay differential equations: the semi-discrete heat equations with time-delay (including two cases, discrete delay and distributed delay) and the neutral differential equations arising from linear circuits of lossless transmission lines.The algorithm decouples a large system into a series of subsystems which can be solved independently; in each iteration the subsystems update the boundary information via "transmission conditions". For the widely used Robin transmission conditions, previous studies show that the convergence factorρof the algorithm satisfies ρ→1 when the mesh size goes to zero, which obviously implies that the convergence rate of the algorithm is not robust with respect to the mesh size h. By using the convolution transmission conditions with optimized tranmission parameterγ,the algorithm possesses strongly robust convergence factor: ρ=1-C,where C∈(0,1) is a mesh independent constant. The transmission parameter γhas a significant effect on the value of the constant C and the best chocie of γdepends,in an extremely complicated relation,on the properties of the delay differential equations and the delay quantity.The goal of this project is to obtain the best delay-dependent transmission parameter and obtain the sharp estimate of the constant C under the best transmission parameter, for the aforementioned several kinds of delay differential equations.

Schwarz 波形松弛是最近十余年提出的一种新型并行计算方法，目前已引起国内外学者广泛关注。本项目进一步深入分析该算法求解几类代表性延迟微分方程时的收敛性：半离散延迟热传导方程（包含离散延迟和分布延迟两种情形），以及描述无损耗传输线电路系统的一类中立型延迟常微分方程组。该算法将一个大规模微分系统拆分成若干可以独立求解的子系统，各子系统在迭代过程中通过“传输条件”更新边界信息。对广泛使用的经典传输条件，前期研究发现：当离散步长趋于0时该算法的收敛因子ρ→1，即算法的收敛速度对离散步长不具有鲁棒性。使用卷积型传输条件并优化传输参数γ可以使算法具有强鲁棒收敛性质：ρ=1-C,其中C∈(0,1)为不依赖于离散步长的常数。参数γ对常数C的大小有显著影响,而最优参数对延迟微分方程的特性及延迟量呈现极复杂的依赖关系。本项目对上述几类方程获得延迟依赖最优传输参数，并获得最优参数下常数C的精确估计。

Schwarz波形松弛算法(简称SWR)是一种新型并行计算方法, 被广泛应用于时间相关偏微分方程和常微分方程组的快速计算中。该算法将一个大规模微分系统拆分成若干可以独立求解的子系统，各子系统在迭代过程中通过“传输条件”更新边界信息。使用卷积型传输条件并优化传输参数γ可以使算法具有强鲁棒收敛性质：ρ=1-C,其中C∈(0,1)为不依赖于离散步长的常数。参数γ对常数C的大小有显著影响,而最优参数对延迟微分方程的特性及延迟量呈现极复杂的依赖关系。项目研究过程中我们对半离散热传导方程以及描述无损耗传输线电路系统的一类中大规模常微分方程组获得了最优传输参数，并获得最优参数下常数C的精确估计。除此之外，我们对SWR算法结合时间并行算法(Parallel-in-Time，简称PinT)进行了大量探索，提出了有效求解波传导问题的ParaDiag算法。该算法具有网格尺寸无关的快速、稳健收敛速度，克服了以Parareal为代表的主流PinT算法求解波传导问题时的本质困难。本项目的研究成果主要以论文形式展现，截止目前，项目组共发表SCI论文16篇（远远超过计划任务书中预期的8-10篇）。