积分微分方程的hp时间步进法及其自适应算法

Integro-differential equations have a very wide application background in many fields of computational scientific and engineering. Due to the existence of nonlocal operators, the theoretical analysis and numerical treatment of these equations are more difficult. The hp time-stepping method has natural advantages for dealing with time-dependent problems. On the one hand, it has high accuracy and is suitable for long-time numerical simulation. On the other hand, it allows variable time steps and variable approximation orders, which is very suitable for adaptive computation. In this project, we shall make full use of the advantages of the hp time-stepping method and focus on several frontier and difficult problems in the field of integro-differential equations, it mainly includes: 1) Consider the Volterra integro-differential equations with weakly singular kernels and random input, develop high accuracy methods in both the parameter space and time direction; 2) Consider the parabolic integro-differential equations, develop an hp version full discretization method with high accuracy in both spatial and temporal directions, and design a reasonable preconditioner to solve the problem effectively; 3) Develop a posteriori error estimation of the hp time-stepping method, and design an efficient adaptive algorithm that can adjust the time steps and the orders of approximation simultaneously, and then apply it to the numerical computation of (partial) integro-differential equations of Volterra type.

积分微分方程在科学和工程计算的诸多领域具有非常广泛的应用背景。由于非局部算子的存在，使得这类方程的理论分析和数值求解更加困难。而hp时间步进法对于处理时间依赖问题具有天然的优势：一方面，它具有高精度，适合长时间数值模拟；另一方面，它允许变时间步长和变逼近次数，非常适合自适应计算。基于此，本项目拟围绕积分微分方程领域的若干前沿和困难问题，充分利用hp时间步进法的优势展开研究，主要包括：1) 针对带有随机输入的弱奇异Volterra积分微分方程，发展随机参数空间和时间方向的双高精度逼近方法；2) 针对抛物型积分微分方程，发展hp型时空全离散高精度方法，并设计合理的预条件子进行高效求解；3) 发展hp时间步进法的后验误差估计理论，设计可同时调节时间步长和逼近次数的高效自适应算法，将其应用于Volterra型（偏）积分微分方程的计算。

积分微分方程在科学和工程计算的诸多领域具有非常广泛的应用背景。本项目针对几类积分微分方程及相关问题的高精度数值方法展开了系统深入的研究，取得了一系列重要成果，主要包括：1）针对具有随机输入的弱奇异Volterra积分微分方程，我们结合随机配置法和hp型时间步进法，构造了随机参数空间和时间方向的双高精度逼近的高效并行算法，并建立了相应的最优误差估计；2）针对具有消失时滞的线性/非线性Volterra积分微分方程，发展了三类高精度的hp型连续Petrov-Galerkin方法、hp型间断Galerkin方法和hp型Chebyshev谱配置法，并分别建立起相应的最优误差估计理论；3）针对弱奇异Volterra积分方程和二阶弱奇异Volterra积分微分方程，分别发展了变阶变步长的hp型间断Galerkin方法和C^0连续Petrov-Galerkin方法，从理论上证明了上述方法对于奇性解逼近的指数型收敛性；4）针对一类广义时滞扩散方程的全离散有限元方法，建立起相应的后验误差估计理论。此外，我们还在与本项目密切相关的领域取得了一些重要进展，譬如：发展了非线性常微分方程和时滞微分方程的hp型Galerkin方法等。项目执行期间，课题组累计在计算数学领域的知名刊物Journal of Scientific Computing、Advances in Computational Mathematics和Communications in Computational Physics等发表SCI论文19篇，基本完成拟定计划和目标。本项目的研究成果进一步发展和丰富了积分微分方程的数值解法，并促进了hp时间步进法在时间依赖问题中的应用。

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