A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel
A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel
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DOI：
10.1016/j.jcp.2020.109576
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发表时间：
2020-09
期刊：
影响因子：
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通讯作者：
Xianming Gu;Shulin Wu
中科院分区：
文献类型：
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作者：
Xianming Gu;Shulin Wu
具有弱奇异核的沃尔泰拉偏积分微分问题由于其在真实的世界中的广泛应用，近年来引起了人们的广泛关注.由于时间演化的非局部性，在时间并行（PinT）模式下求解这类偏微分方程是困难的。在本文中，我们考虑了一类代表性的问题，并提出了一种新的迭代算法PinT计算。在每次迭代中，我们可以同时求解所有离散时间点的偏微分方程，通过最近提出的对角化技术。算法的收敛性进行了分析，洞察到卷积正交权值的递减性质。我们表明，该算法的收敛速度是强大的离散化和问题的参数。数值结果支持我们的研究结果。
Volterra partial integro-differential problems with weakly singular kernel attract a lot of attentions in recent years, thanks to the numerous real world applications. Solving this kind of PDEs in a parallel-in-time (PinT) pattern is difficult, because of thenonlocalproperty of time evolution. In this paper, we consider a class of representative problems and propose a novel iterative algorithm for PinT computation. In each iteration, we can solve the PDEs for all the discrete time points simultaneously via thediagonalizationtechnique proposed recently. Convergence of the algorithm is analyzed by looking insight into the decreasing property of the convolution quadrature weights. We show that the convergence rate of the proposed algorithm is robust with respect to the discretization and problem parameters. Numerical results are reported to support our findings.