Fast wave computation via Fourier integral operators

Abstract

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.