Frozen Gaussian approximation for high frequency wave propagation

Abstract

We propose the frozen Gaussian approximation for computation of high frequency wave propagation. This method approximates the solution to the wave equation by an integral representation. It provides a highly efficient computational tool based on the asymptotic analysis on phase plane. Compared to geometric optics, it provides a valid solution around caustics.Compared to the Gaussian beam method, it overcomes the drawback of beam spreading. We give several numerical examples to verify that the frozen Gaussian approximation performs well in the presence of caustics and when the Gaussian beam spreads. Moreover, it is observed numerically that the frozen Gaussian approximation exhibits better accuracy than the Gaussian beam method. Both authors are grateful to Weinan E for his inspiration and helpful suggestions and discussions that improve the presentation. J.L. would like to thank Lexing Ying for stimulating discussions and for providing preprints [22,23] before publication. Part of the work was done during J.L.'s visits to Institute of Computational Mathematics and Scientific/Engineering Computing of Chinese Academy of Sciences and Peking University, and X.Y.'s visits to Tsinghua University and Peking University. We appreciate their hospitality. We thank anonymous referees for their valuable suggestions and remarks.

Caustic echoes from a Schwarzschild black hole

“…The numerical method must be further improved for very high ratios, possibly using a second order in space formulation [33], implicit-explicit time stepping [34], and adaptive mesh refinement. There are also adapted methods to solve for high-frequency wave propagation, such as the frozen Gaussian beam method [35] or the butterfly algorithm [36], that may improve the efficiency of the numerical simulation considerably.…”

Self-force via Green functions and worldline integration

“…For example, for ultrarelativistic orbits close to the Schwarzschild light ring at r ¼ 3M, the exponentially convergent regime is deferred to very large l [77]. In such cases, it is likely that alternative methods which capture the large-l structure are more suitable, including asymptotic expansions of special functions [24,78], WKB methods [61], the geometrical optics approximation [71], and frozen Gaussian beams [79].…”

“…(7), is smeared out over the entire normal neighborhood by the finite sum over l and contaminates the Green function in that region. A significant gain is therefore possible without decreasing ε or increasing l, but by subtracting the l decomposition of this direct part [24,78] before summing over l. Another approach is to adapt numerical methods for high-frequency wave propagation such as the "frozen Gaussian approximation" [79] based on a paraxial approximation of the wave equation. Yet another is to supplement the numerical approximation of the RGF with (semi-) analytical high-frequency/large-l methods such as the geometrical optics approximation discussed in [71], which was shown to capture the high-frequency behavior of the caustic echoes very accurately, and the large-l asymptotics for the Green function multipolar modes, which accurately capture the global fourfold singularity structure [24,78].…”

A Gaussian Beam Method for High Frequency Solution of Symmetric Hyperbolic Systems with Polarized Waves

“…Specifically, one decomposes the initial wave function into localized wave packets (Gaussian beams) which are then evolved individually along particle trajectories and finally summed up to construct the solution at a later time. It was first studied rigorously in [25], and has seen many recent developments in both Eulerian and Lagrangian frameworks [13,14,15,17,20,21], error estimates [2,19], and fast Gaussian wave decompositions [1,24]. A related approach, known as the Hagedorn wave packet method, was studied in [8,6].…”