A Legendre spectral collocation method-based parabolic equation model for range-dependent acoustic waveguides

Parabolic equation (PE) models are essential tools for simulating underwater sound propagation processes and are important in underwater acoustics research. PE models typically adopt a strategy consisting of vertical discretization followed by horizontal split-step marching, wherein the accuracy and efficiency of the vertical discretization step critically affect the simulation results. Recently, spectral methods have emerged as a new class of high-order accurate numerical discretization techniques and are being increasingly used for vertical discretization in these acoustic models. However, the current Chebyshev–Tau spectrally discretized PE model cannot handle underwater acoustic propagation involving bathymetric variations. In this paper, a wide-angle PE algorithm based on the Legendre spectral collocation method that can simulate waveguides with sloping seabeds is presented. In this algorithm, recursive intermediate variables that satisfy the continuity requirements of spectral discretization are first constructed, and the discontinuous acoustic parameters are then smoothed. For waveguides containing acoustic half-spaces, two sets of Legendre collocation nodes are employed to separately discretize the iterative equations for the water column and the perfectly matched layer. Numerical experiments demonstrate that the proposed Legendre collocation-based PE model can accurately simulate sound propagation processes in waveguides with discontinuous acoustic parameters and varying bathymetries. With respect to waveguides with steep slopes, the Legendre collocation model still accurately captures the corresponding propagation physics without introducing significant errors, thereby verifying the reliability of smoothing discontinuous acoustic parameters and Legendre collocation discretization.