Recovery of High Frequency Wave Fields for the Acoustic Wave Equation
Hailiang Liu, James V. Ralston · Multiscale Modeling and Simulation · 2009
Computation of high frequency solutions to wave equations is important in many applications and notoriously difficult in resolving wave oscillations. Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. An alternative way to compute Gaussian beam components such as phase, amplitude, and Hessian of the phase is to capture them in phase space by solving Liouville-type equations on uniform grids. Following [H. Liu and J. Ralston, Multiscale Model. Simul., to appear] we present a systematic construction of asymptotic high frequency wave fields from computations in phase space for acoustic wave equations; the superposition of phase space based Gaussian beams over two moving domains is shown to be necessary. Moreover, we prove that the kth order Gaussian beam superposition converges to the original wave field in the energy norm at the rate of $\epsilon^{\frac{k}{2}+\frac{1-n}{4}}$ in dimension n.