高振荡薛定谔型方程（组）的高分辨率快速算法

Highly oscillatory Schrödinger-type equations and the related coupled dispersion equations, such as the nonlinear Schrödinger/Gross-Pitaevskii equation, long-wave-short-wave interaction equations and Zakaharov equation in the semi-classical limit, play an important role in Physics. This kind of equations are Hamilton systems in structure, preserve numerous conservative properties in Physics, arise in the practical problems with high-dimensional forms, have highly oscillatory waves solutions. Though several energy conservative methods or symplectic methods were proposed in literature, very few numerical methods can capture successfully the high oscillatory waves. Up to now, establishing the error estimates of the numerical methods with high resolution is still a difficult issue. This program aims to use the spectral method, wavelet method, compact finite difference method, time-splitting method, exponential wave integrator method and multi-scale method to design efficient numerical method with high resolution for solving this kind of equations. Another interest of this program is to establish the optimal error estimates of the proposed schemes with high resolution by using transformation between the time and the space derivatives, cut-off function, introduction argument, H2 technique and the standard energy method.

高振荡薛定谔型方程及与之耦合的高振荡色散方程组，如半经典极限机制下的非线性Schrödinger/Gross-Pitaevskii方程、长短波方程组及Zakaharov方程等，在物理学中占有极为重要的地位。该类方程在物理上满足某些守恒性质，在结构上表现为哈密顿系统，实际问题中往往以高维形式出现，其解更是表现为高频振荡的波形。当前已有一些守恒或保结构算法，但能精确捕捉高频波形的高分辨率快速算法仍极为少见，高分辨率算法的误差估计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、小波方法或紧致差分法、时间方向的指数积分及时空方向的算子分裂和多尺度算法对该类方程构造兼具高效和高分辨率的守恒或保结构算法，并综合运用“时空导数转化”、“Cut-Off函数”和H2估计、归纳论证等技巧结合能量方法对所构造的高分辨率算法建立最优误差估计。

运用有限差分法、Galerkin有限元法、谱和拟谱方法、正交样条配置方法结合算子分裂、时间方向的指数积分等方法对高维非线性广义Schrödinger方程、带有角动量旋转项的Gross-Pitaevskii方程、高维耦合Gross-Pitaevskii方程、Cahn-Hilliard方程、半经典机制下的长短波方程、高维Schrödinger-Boussinesq 方程、N-coupled Schrödinger-Boussinesq 方程、Zakharov-Rubenchik方程、高维耦合Klein-Gordon- Schrödinger方程、高维耦合Klein-Gordon-Zakharov方程等高维非线性方程（组）进行了数值研究，提出了多个高分辨率快速算法，这些算法既具有较高的计算效率，又尽可能多地保持了原方程的某些守恒性质。. 通过引进非线性项的光滑截断技巧、数学归纳技巧和抬升技巧等，结合标准的能量分析方法建立了算法的最优误差估计。特别是对一大类高维非线性偏微分方程（组）的有限差分法、有限元法、拟谱方法、基于积分因子法的运用有限差分法、有限元法、拟谱方法、基于积分因子法的拟谱方法等算法的最优收敛性及收敛阶的分析建立了一个较为成熟的理论分析框架。