Asymptotic stability of a dual-scale compact method for approximating highly oscillatory Helmholtz solutions

Abstract

A novel dual-scale compact method for solving nonparaxial Helmholtz equations at high wavenumbers is proposed and analyzed. The approach is based on decomposing the axisymmetric transverse domain and governing equation according to interconnected micro and macro regions to maintain the smoothness of the underlying problem. Dual compact strategies are then implemented for acquiring highly accurate and efficient beam propagation computations. Aiming at the highly oscillatory solution features, the paper provides a rigorous analysis on the numerical stability. It is shown that the dual-scaled compact method is asymptotically stable. The analysis also reveals necessary constraints for the conventional stability. Computer experiments including self-focusing beam propagation simulations are conducted with various domain scaling factors to validate the theoretical results.