非局部Schrödinger方程的高效守恒算法

In recent years, the non-local differential equations have made rapid progress, and is widely used in such as the anomalous diffusion and rheology. Recently, significant progress appears also in both mathematical analysis and physics applications of non-local Schrödinger equations obtained form the path integral over Levy- process, but there are still very few effective numerical methods. This project studies the efficient conservation numerical methods for a class of non-local Schrödinger equations. Specifically, we consider three problems: 1.For the space semi-discrete non-local Schrödinger initial value problems, we consider the IMEX Runge-Kutta time discretization methods, and such the schemes can conserve the discrete mass and energy, and prove their convergence and stability; 2. We will construct new basis functions which are suitable for the Riesz non-local operator, and use the new basis functions to design the mass-preserving high precision spectral methods for non-local Schrödinger equations; 3. We construct high-order approximate artificial boundary conditions for the non-local Schrödinger equations by the Green functions such that the computational domain and computational complexity can be effectively reduced. The project aims to develop the theory of numerical methods of non-local differential equations and to provide efficient conservation algorithms for the non-local quantum mechanical equations. By numerical calculation and analysis of algorithms, we will reveal the difference between the non-local differential equations and the classical equations, and find out the advantages of non-local equations in the characterization of complex systems.

近年来, 非局部微分方程的研究快速发展, 已获广泛应用, 如反常扩散与流变力学. 最近, 通过对Levy过程路径积分而获得的非局部薛定谔方程在数学分析和物理应用方面都取得了重要进展, 但高效数值方法还很少. 本项目研究非局部薛定谔方程的高效守恒算法. 具体研究三个问题：1. 对于非局部薛定谔方程的空间半离散初值问题, 研究显隐RK时间离散方法, 使得格式能够保持离散质量和能量, 分析算法的稳定性和收敛性. 2. 构造适合非局部Riesz算子的基函数,并以此为基础构造非局部薛定谔方程保质量的空间高精度谱方法. 3. 利用格林函数构造非局部薛定谔方程数值方法的高精度人工边界条件,从而有效的降低计算复杂性和存贮量. 本项目旨在发展非局部方程数值方法的相关理论和为非局部量子力学方程提供高效守恒算法. 通过数值计算和算法分析, 揭示非局部方程和经典方程的差异, 找出非局部方程在刻画复杂系统时的优势.

本项目主要研究具有空间分数阶Laplacian算子的薛定谔方程的高效快速算法。我们主要研究了该方程的时间Strong分裂谱方法，并且严格证明了算法的时间方向的二阶收敛性。我们构造了守恒型差分格式的快速迭代算法，包括HSS快速迭代，PIHSS快速迭代，Krylov子空间方法等，大大提高了计算效率。同时，研究了带有非线性源项的反常扩散方程的二阶BDF方法，证明了算法的收敛性；证明了一类半线性反常扩散方程的耗散性，构造了满足耗散性的数值方法。目前已经发表SCI论文六篇，投稿SCI论文两篇。