周期结构中的非线性亥姆霍兹方程的快速数值算法

项目名称： 周期结构中的非线性亥姆霍兹方程的快速数值算法

项目编号： No.11201508

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： 袁利军

作者单位： 重庆工商大学

项目金额： 22万元

中文摘要： 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。

中文关键词： 非线性亥姆霍兹方程；周期结构；光子晶体；几何光学法；

英文摘要： 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 partial

英文关键词： Nonlinear Helmholtz equation；Periodic structures；Photonic Crystals；Geometrical optics method；