广义薛定谔方程的高精度快速算法及其收敛性分析

Generalized Schr?dinger equations are used widely in quantum physics, this kind of equations are offen high-dimensional nonlinear ones which not only preserve many physical conservation laws such as Charge,Momentum and Energy, but also are Hamilton systems and have symplectic or multi-symplectic structure. A good numerical scheme for solving Schr?dinger equation is not only stable, efficienct and accurate enough but also preserves the structure or some physical conservation laws of the initial problem. In literatures, some instresting accurate and efficient numerical schemes and conservative or structure preserving schemes have been presented to solve Schr?dinger equation, but the schemes which is not only accurate and efficient but also conservative or structure preserving are very few. Especially, the convergence of the accurate and efficient schemes are very difficult to prove.. This item plans to construct some new stable, effient and high-order accurate difference schemes for solving the generalized nonlinear Schr?dinger equation by using spectral method and compact difference in space direction, time-splliting and relaxion in time direction and alternate direction implicit technique, the proposed schemes are also expected to preserve the structure or as many conservative laws as possible. . On theoretical analysis, the item is expected to conquer the difficluty in proving the convergence of the proposed schemes by introducing some new techniques such as "regression of compactness","Cut-Off"technique,"H2"technique and some new energy inequalities; In implemention, this item plans to use some efficient algorithm such as "Thomas algorithm", "algebraic multigrid" method or some parallel algorithms to solve the algebraic system generated from the difference scheme.The proposed schemes are also used to solve some pratical problem such as BEC and super-conductivity.

薛定谔方程在量子物理学中有非常重要的应用，该类方程在物理上满足电荷守恒、能量守恒等多个守恒律，在结构上还是一个Hamilton系统，具有辛和多辛结构，而在形式上多表现为高维非线性。一个好的数值算法不仅要求稳定、快速、足够精确还要尽可能多地保持原问题的结构和某些守恒性质。当前已有一些较为理想的高精度快速算法和能量守恒或保结构算法，但两者兼得的算法尚不多见。特别是高精度快速算法的收敛性证明更是一个亟待解决的难题。本项目拟利用空间方向的谱方法或紧致差分、时间方向的算子分裂、松弛技术、交替方向、本质并行及外推等各种离散手段，对广义非线性薛定谔方程构造稳定的高精度快速算法，同时要求算法尽可能多的保持原问题的结构和某些守恒性质。理论上，引入'紧致性回归'、"Cut-Off"和H2技巧等一些新的能量分析手段来证明算法的稳定性和收敛性。计算上，运用代数多重网格法或某些并行技术对离散后的代数方程组进行求解。

该项目对几类广义薛定谔方程进行了数值研究，构造了若干个高精度快速算法并对算法的守恒性、稳定性及收敛性进行了分析。对非线性薛定谔方程构造了一类紧致差分格式，首次证明该格式也能保持总能量守恒，并引入张量积、矩阵分析、Cut-off 函数、归纳论证等技巧，运用能量方法建立了数值解对精确解的最优误差估计。随后，该算法及相关分析技巧成功运用到耦合Schrodinger方程、Klein-Gordon-Schrodinger方程、高维Gross-Pitaevskii方程等保持总能量守恒的非线性色散方程组的数值求解中，得到了非常理想的计算结果。对高维耦合非线性Schrodinger方程组运用分裂谱方法进行了数值研究，构造了一个时间方向具有4阶精度，空间方向具有谱精度的稳定地显式数值算法，与有限差分算法和有限元算法，该算法在对高维耦合非线性Schrodinger方程的计算中在精度和效率方面都有不可比拟的优势。对高振荡的几个色散方程的紧致差分算法进行了分析，严格分析了网格步长的选取对振荡参数的依赖情况，指出了有限差分法在高振荡问题的数值求解上有本质局限性，并提出了一个对网格选取较为宽松的数值算法。我们还对分数阶方程和非线性耦合Schrodinger方程的平面动力系统进行了数学和数值分析，得到了一些好的结果。