Fast wave computation via Fourier integral operators
Fast wave computation via Fourier integral operators
复制标题
DOI：
10.1090/s0025-5718-2012-02557-9
复制
发表时间：
2012-09
期刊：
影响因子：
--
通讯作者：
L. Demanet;Lexing Ying
中科院分区：
文献类型：
--
作者：
L. Demanet;Lexing Ying
本文提出了求解“时间上尺度”波动方程的一种数值方法，即执行不受CFL条件限制的时间步长。提出的方法利用了最近在伪微分和傅立叶积分算子（FIO）的快速算法方面的工作。这种算法方法不是渐近的：它展示了如何通过1)求解相位的Hamilton-Jacobi方程，以及2)随机采样低秩矩阵的行和列来构造精确的FIO传播子。我们感兴趣的设置是二维平滑周期介质中的标量波（环面上的c1类），其中波的带宽限制N趋于无穷。在这种情况下，证明了将波动方程求解到固定时间T ' 1的算法复杂度可以低至O(n2 logN)，并且精度可控。数值实验表明，在某些有物理意义的情况下，该方法的时间复杂度比光谱方法低。
This paper presents a numerical method for \time upscaling" wave equations, i.e., performing time steps not limited by the Courant-Friedrichs-Lewy (CFL) condition. The proposed method leverages recent work on fast algorithms for pseudodierential and Fourier integral operators (FIO). This algorithmic approach is not asymptotic: it is shown how to construct an exact FIO propagator by 1) solving Hamilton-Jacobi equations for the phases, and 2) sampling rows and columns of low-rank matrices at random for the amplitudes. The setting of interest is that of scalar waves in two-dimensional smooth periodic media (of class C 1 over the torus), where the bandlimit N of the waves goes to innity. In this setting, it is demonstrated that the algorithmic complexity for solving the wave equation to xed time T ’ 1 can be as low as O(N 2 logN) with controlled accuracy. Numerical experiments show that the time complexity can be lower than that of a spectral method in certain situations of physical interest.