Galerkin spectral-element methods for parabolic equations in underwater acoustics with range-dependent environments

Accurate modeling of sound propagation is vital for underwater acoustics in ocean exploration and communication. Classical wide-angle parabolic equation models, often discretized using low-order finite difference or finite element schemes, face limitations in accuracy and efficiency. While the Chebyshev-Tau spectral method based SMPE improves accuracy, it requires constant layer thickness during forward stepping, making it unsuitable for range-dependent seafloor problems. To address these limitations, we propose a Galerkin spectral-element methods for solving split-step Padé energy-conserving parabolic equations. Our method relaxes the regularity requirements of solutions, enhances complex boundary condition handling, and generates a symmetric, block-diagonal matrix system, improving computational efficiency. Numerical experiments demonstrate the method’s accuracy and performance, particularly in up-slope propagation simulations in wedge-shaped oceans, where it achieves high-quality results with fewer depth interpolation points and larger range-step sizes. This approach offers a robust and efficient solution for range-dependent underwater acoustic modeling.