周期结构中的非线性亥姆霍兹方程的快速数值算法

The Helmholtz equation is the governing equations for various of waves such as sound waves, electromagnetic waves and light. When waves propagating in nonlinenar media, the governing equations become a nonlinear Helmholtz equation. The nonlinear properties of wave propagating in nonlinear media have many applications, such as the application in optical fiber, ultra-short wave laser generation and biomedical imaging. Therefore, it is important to develop efficient numerical simulation methods for solving the nonlinear Helmholtz equations. Due to the fact that a discretization of the linear Helmholtz equation gives rise to a complex, non-Hermitian, non-diagonally dominant and highly indefinite system, it is difficult to even solve the linear Helmholtz equation in a large domain by exsiting iterative methods. However, our previous research works show that when the media have some useful geometric features, such as periodicity, it is possible to develop more efficient numerical methods for linear Helmholtz equation. The nonlinear Helmholtz equations are more difficult to solve, because the iterative method for solving nonlinear Helmholtz equation may have a slow convergence or fail to converge. In this project, we develop some fast numerical methods for solving the nonlinear Helmholtz equation in periodic or partially periodic structures by taking advantages of the geometric features of the problem.

亥姆霍兹（Helmholtz）方程可以描述各种波在介质中的传播，例如声波、电磁波和光波。当波在非线性介质中传播时，其控制方程是非线性Helmholtz方程或方程组。波传播的非线性特性具有很多有趣的现象和应用，例如可以用来提高光纤的传播性能、超短激光的产生、生物医学成像等。因而开发求解非线性Helmholtz方程或方程组的高效算法非常重要。由于Helmholtz方程被离散后，通常会得到一个复的、不定的系数矩阵，所以即使要快速求解大型结构的线性Helmholtz方程依然很困难。然而，当所求结构具有某种有用的几何特性时（例如周期性），可以开发出求解线性Helmholtz方程的快速算法。求解非线性Helmholtz方程更困难，因为一般的求解非线性方程的迭代方法收敛很慢，或者不收敛。在本项目中，我们将利用结构的几何特性，开发求解周期或部分周期结构中的非线性Helmholtz方程或方程组的快速算法。

非线性亥姆霍兹方程可描述光波在非线性介质中的传播。非线性光波在现代科学技术中有很重要的应用，例如可以用来提高光纤的传播性能、超短激光的产生、生物医学成像等。非线性亥姆霍兹方程边值与特征值问题的理论研究与数值研究受到了国内外学者的普遍重视。快速求解一般结构中的非线性亥姆霍兹方程非常困难。通过本项目的实施，我们开发了求解周期或部分周期结构中具有Kerr非线性的亥姆霍兹方程的快速数值算法。并利用此方法求解不同结构中的非线性亥姆霍兹方程的边值与特征值问题，研究了对称性破缺、双稳定性和非线性驻波等非线性现象。此外，本项目还开发了基于模式展开法的近似快速求解线性三维麦克斯韦方程组的数值方法，在较少的误差下将计算速度提高了至少27倍；开发了基于Babich展开的几何光学法求解高频二维与三维亥姆霍兹方程的点源问题，由于Babich展开的振幅函数在源点附近是光滑的，此方法更容易实现。