几类非局部微分方程的高效保结构算法

The nonlocal differential equations including especially the fractional-order differential equations are better able to describe many complex processes and phenomena, and its modeling, theory, computation and application have become a hot research topic in many fields. This kind of equations is of nonlocal characteristics: long history memory in time or long-range interactions in space. At present, the research into their numerical methods focus on high-precision algorithms and their theory, and the improvement of computational efficiency by reducing the large amount of calculation and storage. Many researches show that some nonlocal differential equations are of some conservation quantities (such as mass quality, energy) or asymptotic behaviors (such as attractor, contractility, dissipativity), but the structure-preserving algorithms for this class of equations are only a small amount of scattered researches at home and abroad. The project will apply the structure-preserving idea into the construct of numerical methods for nonlocal differential equations, and overcome some essential difficulties caused by some complex factors such as the time or space non-locality. By paying the same attentions to structure-preserving as accuracy and efficiency, we will research mainly efficient structure-preserving difference methods and their theory for initial (boundary) value problems of time-nonlocal Volterra functional differential equations (VFDEs) including especially the fractional-order VFDEs, and space-nonlocal Schrödinger equations. Moreover, these algorithms and their theory are extended to other methods and the computation of some related problems. These will enrich and develop the structure preserving algorithms and their theory of nonlocal differential equations, and provide useful help to the related applications.

分数阶微分方程等非局部微分方程能更好地描述许多复杂过程和现象，其建模、理论、计算、应用已成为众多领域的研究热点。此类方程具有非局部特征：长期历史记忆性或空间全域相关性，其数值方法研究目前主要关注：高精度算法构造及其理论、减少大计算量和存储量以提高计算效率。许多研究表明：此类方程具有质量、能量等守恒量或吸引子、收缩性、耗散性等渐近性态，但目前国内外关于其保结构算法仅有少量零散研究。本项目将保结构思想应用于此类方程数值方法的构造，克服时间或空间非局部性等复杂因素所带来的实质困难，同时关注算法的保结构性、精度和效率，主要研究时间非局部Volterra泛函微分方程(尤其是分数阶Volterra泛函微分方程)、空间非局部Schrödinger方程初（边）值问题的高效保结构差分方法及其理论，并拓广到一些其它方法和相关问题计算，丰富和发展非局部微分方程的保结构算法及其理论，为相关应用提供有益的帮助。

非局部问题常出现于具有长时间记忆性或长程相关性的科学、工程、社会、经济金融问题中，深入研究其理论和数值方法具有重要的科学意义和广泛的应用前景。项目研究主要包括：三类非局部微分方程的结构性质，时间非局部Volterra泛函微分方程、空间分数阶Schrödinger方程及相关问题的高效（保结构）方法，基于偏积微分方程的带跳期权定价问题、分数阶扩散方程及随机、延迟、刚性微分方程的高效算法及其理论。. 项目获许多有意义的新性质、新结果、新高效（保结构）算法，主要包括：1）通过建立新分数Halanay型不等式，在与整数阶情形相同的假设下获时间分数阶中立型泛函微分方程解的多项式收缩性和耗散性。2）通过建立离散分数阶能量不等式和离散Volterra差分方程长时间最优估计，证明时间分数阶非线性系统BDF公式保持连续解的长时间多项式耗散率和收缩率。3）针对Levy噪声驱动的带双奇异核的随机Volterra积分方程和带正则核的非Lipschitz随机分数阶积微分方程，建立解的适定性，分别构造快速、修正的Euler-Maruyama方法，获强收敛结果；另实质改进分数噪声驱动的过阻尼广义Langevin方程Euler法的已有强收敛结果。4）构造1维空间分数阶Schrödinger方程的4阶辛差分格式及空间分数阶量子修正Zakharov方程的线性隐式守恒差分格式。5）发展时间分布阶扩散方程的有限差分-有限元计算框架、三类组合格式与二维回火时间分数阶扩散方程的高效差分-有限元格式及其收敛理论。6）提出1维Riemann-Liouville分数阶微分方程的高效h-p Petrov-Galerkin有限元方法。7）对基于偏积微分方程的带切换跳期权定价问题，建立隐显BDF2时间离散方法及L2稳定性。8）针对一些多重随机积分，找到其均方渐近最优逼近，相比目前主要逼近方法更高效。9）首次获延迟扩散方程的长时间后验误差估计。10）发展刚性随机微分方程的新显式稳定随机Runge-Kutta方法及刚性系统的改进的Runge-Kutta-Chebyshev方法。. 项目成果丰富和发展了非局部、随机、记忆微分方程数值方法的研究，可为相关实际问题的计算服务。部分成果获湖南省自然科学奖二等奖、陕西省青年科技奖、中国仿真学会优秀博士论文奖等。