A Hybrid Algorithm Based on Optimal Quadratic Spline Collocation and Parareal Deferred Correction for Parabolic PDEs

Liu Jun; Wang Yan; Li Rongjian

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A Hybrid Algorithm Based on Optimal Quadratic Spline Collocation and Parareal Deferred Correction for Parabolic PDEs

机译：基于最优二次样条搭配和抛物线延迟校正的抛物线偏微分方程混合算法

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Parareal is a kind of time parallel numerical methods for time-dependent systems. In this paper, we consider a general linear parabolic PDE, use optimal quadratic spline collocation (QSC) method for the space discretization, and proceed with the parareal technique on the time domain. Meanwhile, deferred correction technique is also used to improve the accuracy during the iterations. In fact, the optimal QSC method is a correction of general QSC method. Along the temporal direction we embed the iterations of deferred correction into parareal to construct a hybrid method, parareal deferred correction (PDC) method. The error estimation is presented and the stability is analyzed. To save computational cost, we find out a simple way to balance the two kinds of iterations as much as possible. We also argue that the hybrid algorithm has better system efficiency and costs less running time. Numerical experiments by multicore computers are attached to exhibit the effectiveness of the hybrid algorithm.

机译：Parareal是一种用于时间相关系统的时间并行数值方法。在本文中，我们考虑了一般的线性抛物线PDE，使用最佳二次样条搭配（QSC）方法进行空间离散化，并在时域上进行了超现实技术。同时，还使用了延迟校正技术来提高迭代过程中的精度。实际上，最佳QSC方法是对一般QSC方法的修正。沿着时间方向，我们将延迟校正的迭代嵌入到parareal中，以构造混合方法，即parareal延迟校正（PDC）方法。给出了误差估计并分析了稳定性。为了节省计算成本，我们找到了一种尽可能平衡两种迭代的简单方法。我们还认为，混合算法具有更好的系统效率和更少的运行时间。附加了多核计算机的数值实验，以展示混合算法的有效性。

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《Mathematical Problems in Engineering》 |2016年第5期|6943079.1-6943079.15|共15页
作者
Liu Jun; Wang Yan; Li Rongjian;
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China Univ Petr East China, Coll Sci, Qingdao 266580, Shandong, Peoples R China;

China Univ Petr East China, Coll Sci, Qingdao 266580, Shandong, Peoples R China;

Old Dominion Univ, Dept Comp Sci, Norfolk, VA 23508 USA;

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1. An enhanced parareal algorithm based on the deferred correction methods for a stiff system [J] . Sunyoung Bu, June-Yub Lee Journal of Computational and Applied Mathematics . 2014,第Null期

机译：基于延迟校正方法的刚性系统增强型Parareal算法
2. ON PARAREAL ALGORITHMS FOR SEMILINEAR PARABOLIC STOCHASTIC PDEs [J] . Brehier Charles-Edouard, Wang Xu SIAM Journal on Numerical Analysis . 2020,第1期

机译：关于半线性抛物线随机PDE的宫算法
3. High order numerical algorithms based on biquadratic spline collocation for two-dimensional parabolic partial differential equations [J] . Liu Jun, Guo Hui, Jiang Yao-Lin, International journal of computer mathematics . 2019,第1a4期

机译：二维抛物型偏微分方程的基于二次二次样条配点的高阶数值算法
4. Quadratic Spline Collocation and Parareal Deferred Correction Method for Parabolic PDEs [C] . Jun Liu, Yan Wang, Rongjian Li International Conference of Numerical Analysis and Applied Mathematics . 2016

机译：抛物线PDE的二次样条搭配和窥柱渗透校正方法
5. An efficient algorithm based on quadratic spline collocation and finite difference methods for parabolic partial differential equations. [D] . Chen, Tong. 2005

机译：一种基于二次样条搭配和有限差分法的抛物型偏微分方程的高效算法。
6. Numerical Algorithm Based on Haar-Sinc Collocation Method for Solving the Hyperbolic PDEs [O] . A. Pirkhedri, H. H. S. Javadi, H. R. Navidi -1

机译：基于Haar-Sinc配置方法的双曲型偏微分方程数值算法
7. Parareal and Spectral Deferred Corrections [O] . Michael L. Minion, Sarah A. Williams 2010

机译：超现实和光谱延迟校正

A Hybrid Algorithm Based on Optimal Quadratic Spline Collocation and Parareal Deferred Correction for Parabolic PDEs

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