Error estimates for Gaussian beam superpositions

Abstract

Abstract. Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations, concentrated on a single curve through the physical domain. They can also be extended to some dispersive wave equations, such as the Schrödinger equation. Superpositions of Gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve. This work is concerned with the accuracy of Gaussian beam superpositions in terms of the wavelength ε. We present a systematic construction of Gaussian beam superpositions for all strictly hyperbolic and Schrödinger equations subject to highly oscillatory initial data of the form Ae iΦ/ε . Through a careful estimate of an oscillatory integral operator, we prove that the k-th order Gaussian beam superposition converges to the original wave field at a rate proportional to ε k/2 in the appropriate norm dictated by the well-posedness estimate. In particular, we prove that the Gaussian beam superposition converges at this rate for the acoustic wave equation in the standard, ε-scaled, energy norm and for the Schrödinger equation in the L 2 norm. The obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics. We present a numerical study of convergence for the constant coefficient acoustic wave equation in R 2 to analyze the sharpness of the theoretical results.

Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

“…This result is proven in, e.g., [7]. A general initial condition can be approximated by an integral superposition of Gaussian beams, also then the propagation error is bounded by C T ε 1/2 [21]. In [31, Theorem 2.1] it was shown how to approximate a smooth function by an integral superposition of Gaussian beams, and what error bounds such superpositions satisfy.…”

“…By adding higher order terms to the amplitude and phase, higher order Gaussian beams can be constructed. Error estimates, in terms of the parameter ε, for more general potentials and Gaussian beams of arbitrary order were shown in [21]. In parallel to their development in the applied mathematics community, Gaussian beams were discovered by chemical physicists [8] motivated by the observation that if the potential is a quadratic polynomial, a Gaussian wave function will stay Gaussian for all time.…”

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

“…Convergence results for the decomposition of the initial condition are presented in [19,28]; here we derive improved convergence results. The final summing process for Eulerian Gaussian beams was shown to lose accuracy after the formation of caustics in [13]; here we introduce a new Eulerian summation formula to solve this problem.…”

“…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].…”

Gaussian beam methods for the Dirac equation in the semi-classical regime

“…The validity of the Gaussian beam method at caustics was analyzed by Ralston in [28]. The Gaussian beam and related methods have become very popular in high frequency waves problems [18,19,20,23,27,33,34,39,40] in recent years. Most of the methods were in the Lagrangian framework.…”