广义薛定谔方程的高精度快速算法及其收敛性分析

项目名称： 广义薛定谔方程的高精度快速算法及其收敛性分析

项目编号： No.11201239

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 王廷春

作者单位： 南京信息工程大学

项目金额： 22万元

中文摘要： 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。

中文关键词： 非线性Schrodinger方程；高精度快速算法；守恒律；稳定性；误差估计

英文摘要： 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 theo

英文关键词： Nonlinear Schrodinger equation；Accurate and efficient numerical method；Conservative laws；Stability；Error estimate