Fast Multiscale Gaussian Wavepacket Transforms and Multiscale Gaussian Beams for the Wave Equation

We introduce a new multiscale Gaussian beam method for the numerical solution of the wave equation with smooth variable coefficients. The first computational question addressed in this paper is how to generate a Gaussian beam representation from general initial conditions for the wave equation. We propose fast multiscale Gaussian wavepacket transforms and introduce a highly efficient algorithm for generating the multiscale beam representation for a general initial condition. Starting from this multiscale decomposition of initial data, we propose the multiscale Gaussian beam method for solving the wave equation. The second question is how to perform long time propagation. Based on this new initialization algorithm, we utilize a simple reinitialization procedure that regenerates the beam representation when the beams become too wide. Numerical results in one, two, and three dimensions illustrate the properties of the proposed algorithm. The methodology can be readily generalized to treat other wave propagation problems.