A fourth-order numerical method for the one-dimensional nonlinear Helmholtz equation with multilayered material

Abstract

Based on a compact approximation of finite element type, a fourth-order numerical scheme is developed for the one-dimensional nonlinear Helmholtz equation with multilayered material. The Hermite–Birkhoff interpolation is employed to approximate the field between the grid nodes with fourth-order accuracy. The second-order derivatives that appear in the Hermite–Birkhoff interpolation are evaluated by the original differential equation. The resulting nonlinear system of equations is solved by using the Newton’s iterations combined with a continuation method. Numerical experiments are conducted to demonstrate that the scheme possesses the anticipated order of accuracy. To validate the numerical method, we compute several numerical experiments about optical bistability phenomenon with the developed scheme.