Asymptotic solutions for high frequency Helmholtz equations

In this thesis, we will investigate and develop asymptotic methods for numerically solving high frequency Helmholtz equations with point-source conditions. Due to the oscillatory nature of the wave, such equations are highly challenging to solve by conventional methods, such as the finite difference and finite element methods, since they often suffer from a large number of degrees of freedom to avoid the `pollution effect\u27 (large dispersion errors). We shall first apply the geometrical optics (GO) approximation to compute the wave locally near the primary source, where instead of computing the oscillatory wave directly, its phase and amplitudes are computed through the eikonal equation and a recurrent system of transport equations, respectively, and are used to reconstruct the wave for any high frequencies. The GO approximation is efficient for providing locally valid approximations of the wave. We propose to further propagate the wave to the whole domain of interest through an appropriate time-dependent Schr\ {o}dinger equation whose steady-state solution in the domain of interest will provide globally valid approximations of the wave. The wavefunction of the Schr\ {o}dinger equation can be propagated by a Strang operator splitting based pseudo-spectral method that is unconditionally stable, which allows large time step sizes to reach the steady state efficiently. In the pseudo-spectral method, wherever the matrix exponential is involved, the Krylov subspace method can be used to compute the relevant matrix-vector products. The proposed asymptotic method will be effective since: (1) it is able to obtain globally valid approximations of the wave, (2) it has complexity $O(N\log N)$ where $N$ is the total number of simulation points for a prescribed accuracy requirement, and (3) the number of simulation points per wavelength can be fixed as the frequency increases. Numerical experiments in both two- and three-dimensional spaces will be performed to demonstrate the method