复非线性问题的高效快速数值算法研究及应用

Researches on numerical solution of nonlinear systems have significance for natural sciences and engineering. This project is considered the complex nonlinear Schrödinger and Ginzburg-Landau equations as the breakthrough point, which obtained from nonlinear optics, condensed matter physics and other practical problems, focusing on the cases of one-dimensional, high-dimensional and fractional order, we try to explore the efficient and fast numerical algorithms for solving the complex nonlinear equations based on different discrete methods. First, we analyze the inherent structural characteristics of such complex nonlinear systems. We explore and propose new nonlinear iterative methods based on the idea of multi-step iteration, over relaxation, Anderson acceleration and global strategy. Secondly, according to the different nature of the linear system obtained by nonlinear iterative method for solving the complex system, considering the structural characteristics of the Jacobi matrix, using the idea of matrix splitting and alternating direction, we aim to establish new numerical algorithms for solving the complex linear systems, and employ them as the inner iterative method for solving complex nonlinear equations. The precondition technology is considered to improve speed of numerical solution. Finally, the internal influence and relationship between the inner iteration method and the outer iteration method are analyzed. And we aim to establish the local and semilocal convergence theorems, besides, we intend to construct the basic framework of convergence theory.

非线性方程组数值算法研究对科学和工程都有重要意义。本项目以源于非线性光学、凝聚态物理等实际问题的复非线性Schrödinger、Ginzburg-Landau方程为切入点，着眼于该类模型的一维、高维及分数阶情形，探索这几类方程根据不同离散方法得到的复非线性方程组的高效快速数值算法研究。首先分析这类复非线性系统的内在结构特征，借助多步迭代法思想、超松弛思想、Anderson加速、全局策略等，探索和提出新的非线性迭代法。其次，针对由非线性迭代法求解中得到的不同性质的线性复子系统，考虑Jacobi矩阵的结构特征，利用矩阵分裂、方向交替思想等，探索建立新的求解复线性系统的数值算法，并以此作为求解复非线性方程组的内迭代算法。考虑高效预处理子的改进与构造，提升数值求解效率。最后，理论上研究所构造内外迭代方法，分析内迭代法与外迭代法的内在影响与关系，建立局部、半局部收敛性定理，构建收敛理论的基本框架。

非线性Schrödinger、Ginzburg-Landau方程是源于非线性光学和流体力学等诸多物理分支和应用数学的重要的非线性数学模型，这类数学模型离散后得到的非线性方程组，其Jacobi矩阵具有复系数的代数结构。本项目针对复非线性方程组的高效数值算法进行了如下几方面的研究：. 1.基于对复非线性方程由不同方法离散后的复非线性系统的结构特性分析，考虑影响迭代收敛的因素，我们研究了新的非线性迭代算法的构造，我们分别针对一般复非线性方程组，块结构复非线性方程组，弱复非线性方程组等问题，充分考虑了复非线性方程组的结构问题，设计了以高阶Newton型方法以及Picard型方法为外迭代，以高效线性迭代法为内迭代的快速内外迭代算法。. 2.对非线性迭代法求解过程中产生的线性子问题，我们考虑了系数矩阵的复线性对称性，块复线性结构，高维性等，建立了新的求解线性系统的数值方法，给出了理论分析。特别我们将块递推算法推广到具有Kronecker乘积结构的线性系统的求解。为了提高算法效率，我们引入了维数合并技巧，我们也给出了所构造算法的复杂性分析。. 3.研究了线性迭代法的收敛性在作为求解非线性问题的内迭代对整体算法的影响与关系。理论上分析了所构造的内外迭代数值算法，研究了其局部、半局部等收敛性，建立收敛性定理。. 本项目在执行期取得了一些相关成果，发表了标注资助名称和项目编号SCI学术检索论文7篇。