A Convergent Multiscale Gaussian-Beam Parametrix for the Wave Equation

The Gaussian beam method is an asymptotic method for wave equations with highly oscillatory data. In a recent published paper by two of the authors, a multiscale Gaussian beam method was first proposed for wave equations by utilizing the parabolic scaling principle and multiscale Gaussian wavepacket transforms, and numerical examples there demonstrated excellent performance of the multiscale Gaussian beam method. This article is concerned with the important convergence properties of this multiscale method. Specifically, the following results are established. If the Cauchy data are in the form of non-truncated multiscale Gaussian wavepackets, the multiscale Gaussian beam method provides a convergent parametrix for the wave equation with highly oscillatory data, and the convergence rate is, where 1/√λ, is the smallest frequency contained in the highly oscillatory data. If the highly oscillatory Cauchy data are in the form of truncated multiscale Gaussian wavepackets, the multiscale Gaussian beam method converges with a rate controlled by, where 1/√λ+ε is the error from initializing the Gaussian beam method by multiscale Gaussian wavepacket transforms. To prove these convergence results, it is essential to characterize multiscale properties of wavepacket interaction and beam decaying by carrying out some highly-oscillatory integrals of Fourier-integral-operator type, so that those multiscale properties lead to precise convergence orders for the multiscale Gaussian beam method.