Fast Huygens sweeping methods for Helmholtz equations in inhomogeneous media in the high frequency regime
Fast Huygens sweeping methods for Helmholtz equations in inhomogeneous media in the high frequency regime
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高频区域非均匀介质中亥姆霍兹方程的快速惠更斯扫描方法
DOI：
10.1016/j.jcp.2014.03.066
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
R. Burridge
中科院分区：
文献类型：
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作者：
S. Luo;J. Qian;R. Burridge
在某些应用中，假设测地线（射线）具有一致的方向是合理的，因此亥姆霍兹方程可以被视为一个空间方向上的演化方程。考虑到这样的应用，我们提出了一个新的欧拉计算几何光学方法，称为快速惠更斯扫描法，计算绿色函数的亥姆霍兹方程在非均匀介质中的高频制度，并在焦散线的存在。新方法的第一个新奇是，惠更斯-基尔霍夫二次源原理是用来集成许多局部有效的渐近解，以产生一个全球有效的渐近解，使焦散与通常的几何光学测量可以自动处理。第二个新奇在于，蝶形算法适于在O（N log N）操作中执行由Huygens-Kirchhoff积分引起的矩阵-向量乘积，其中N是网格点的总数，并且比例常数取决于所需的精度并且与频率参数无关。为了减少存储所得到的走时和振幅表，我们压缩每个表成基于张量积的多元切比雪夫多项式的线性组合，使每个表的信息被编码成少量的切比雪夫系数。新方法具有以下特点：（1）可预先计算一组局部走时和振幅表;（2）可自动处理焦散线;（3）可构造任意频率和多点源的亥姆霍兹方程的绿色函数;（4）对于每个波长的指定数量的点，它以网格点的总数方面的接近最优的复杂度构造每个绿色函数，其中复杂度的前因子仅取决于指定的准确度并且与频率参数无关。两个二维（2-D）和三维（3-D）的数值实验，以证明新方法的性能和精度。
In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian computational geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green functions of Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the new method is that the Huygens–Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The second novelty is that a butterfly algorithm is adapted to carry out the matrix–vector products induced by the Huygens–Kirchhoff integration in O (N log N) operations, where N is the total number of mesh points, and the proportionality constant depends on the desired accuracy and is independent of the frequency parameter. To reduce the storage of the resulting traveltime and amplitude tables, we compress each table into a linear combination of tensor-product based multivariate Chebyshev polynomials so that the information of each table is encoded into a small number of Chebyshev coefficients. The new method enjoys the following desired features:(1) it precomputes a set of local traveltime and amplitude tables;(2) it automatically takes care of caustics;(3) it constructs Green functions of the Helmholtz equation for arbitrary frequencies and for many point sources;(4) for a specified number of points per wavelength it constructs each Green function in nearly optimal complexity in terms of the total number of mesh points, where the prefactor of the complexity only depends on the specified accuracy and is independent of the frequency parameter. Both two-dimensional (2-D) and three-dimensional (3-D) numerical experiments are presented to demonstrate the performance and accuracy of the new method.