基于非均匀网格的Helmholtz方程的优化差分法及其预处理迭代算法

项目名称： 基于非均匀网格的Helmholtz方程的优化差分法及其预处理迭代算法

项目编号： No.11301310

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 吴亭亭

作者单位： 山东师范大学

项目金额： 23万元

中文摘要： Helmholtz 方程在地球物理等领域中有重要应用，研究其高性能数值解法具有重要的理论意义和应用价值。差分法是油气勘探中一种常用的数值方法。均匀网格差分法求解该方程时，为保证计算精度，须使用由低速度所决定的较小的网格步长对整个区域进行离散，对大部分多尺度模型而言，将会导致计算量的增加和计算机资源的浪费。针对这一困难，本项目拟综合应用不连续和连续的非均匀网格技术对带完美匹配层（PML）的计算区域进行网格剖分；利用虚拟差分点或加权平均的思想构造差分格式；拟提出推广化的加细选取策略来选取优化系数，最终建立与带PML的Helmholtz方程相容的优化差分格式；拟提出一种新的限制和插值算子来构造依赖于矩阵的多重网格法，并采用多重网格预处理的Krylov子空间方法来求解离散化得到的线性系统。研究成果将提高差分法对模型的适应性，降低对内存的需求，减少计算时间，为大规模实际问题的计算奠定坚实的基础。

中文关键词： Helmholtz 方程；完美匹配层；非均匀网格；有限差分法；多重网格

英文摘要： The Helmholtz equation has many applications in geophysics. To develop high performance numerical methods for solving this equation is of important theoretical significance and great application value. The finite difference scheme is a popular method in the oil exploration. When using uniform grids, for the purpose of accuracy, we must use the grid size established by the slowest velocity to discretize a model. As a result, heterogeneous models are always oversampled, which leads to unnecessary extra computations. To overcome this difficulty, we will combine the discontinuous nonuniform grids and the continuous nonuniform grids to discretize the computation domain and its perfectly matched layer (PML). Then, we will construct the finite difference equations by the ghost points or the weighted-averaging difference operators, and we will propose a generalized refined strategy for choosing optimal parameters of the finite difference scheme. Finally, we formulate the consistent and optimal finite difference methods for the Helmholtz equation with the PML on the nonuniform grids. A new matrix-based restriction and interpolation operator will be proposed for the multigrid method, and a multigrid-based preconditioned Krylov subspace method will be fromulated for the linear systems after discretization. The researches

英文关键词： the Helmholtz equation；the perfectly matched layer；nonuniform grids；The finite difference scheme；the multigrid