Fast wave computation via Fourier integral operators
Author(s)Demanet, Laurent; Ying, Lexing
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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 pseudodifferential 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∞ over the torus), where the bandlimit $ N$ of the waves goes to infinity. In this setting, it is demonstrated that the algorithmic complexity for solving the wave equation to fixed time T ≃ 1 can be as low as O(N [superscript 2] log N) 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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Mathematics of Computation
American Mathematical Society (AMS)
Demanet, Laurent, and Lexing Ying. “Fast Wave Computation via Fourier Integral Operators.” Mathematics of Computation 81.279 (2012): 1455–1486. Web. 11 Apr. 2012.
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