Numerical solutions for point-source high frequency Helmholtz equation through efficient time propagators for Schrödinger equation

Abstract

For high frequency Helmholtz equations with point-source conditions, geometrical optics provides an efficient way to compute the solutions without ‘pollution effect’ that is usually unavoidable in finite difference and finite element methods. However, geometrical optics generally can only provide locally valid approximations for the wavefield near the primary source. In order to obtain globally valid wavefield, we will propagate the wavefield with efficient time propagators for an appropriate time-dependent Schrödinger equation. The geometrical optics is first applied to compute locally valid approximations for the wavefield, then the wavefield is propagated through the time-dependent Schrödinger equation until its change is small enough in the domain of interest, which results in the globally valid approximation for the solution to the Helmholtz equation in the domain of interest. The adapted time propagator for the time-dependent Schrödinger equation is a Strang operator splitting based pseudospectral approach, which assures the overall efficiency of the method since the pseudospectral approach is unconditionally stable such that large time step sizes are allowed to propagate the wavefield efficiently, and the number of points per wavelength is independent of the frequency. Both two-dimensional and three-dimensional numerical experiments are presented to demonstrate the method.