Exponential integrator method for solving the nonlinear Helmholtz equation

In this paper, we study the exponential integrator method (EIM) for solving the nonlinear Helmholtz equation (NLHE). As the wave number or the characteristic coefficient in the nonlinear term is large, the NLHE becomes a highly oscillatory and indefinite nonlinear problem, which makes most of numerical methods lose their expected computational effects. Based on the shooting method, the NLHE is firstly transformed into an initial-value-type problem. Then, the EIM is utilized for solving the deduced problem, by which we not only can capture the oscillation very well, but also avoid to search the nonlinear iteration method and to solve indefinite linear equations at each iteration step. Therefore, the high accuracy simulations with relative large physical parameters in the NLHE become possible and lots of computational costs can be saved. Some numerical examples, including the extension to the nonlinear Helmholtz system, are shown to verify the accuracy and efficiency of the proposed method.