Three-dimensional Cartesian parabolic equation model with higher-order cross-terms using operator splitting, rational filtering, and split-step Padé algorithm.

An approximate form of three-dimensional Cartesian split-step marching solution for the acoustic parabolic equation is derived in order to obtain the efficient algorithm for sound propagation in the three-dimensional ocean. The operator splitting method is used to split the full exponential operator into three exponential operators for depth, cross-range, and the combination of the two. The first two terms are implemented with the split-step Padé algorithm and the final term is implemented with the Taylor series expansion in depth and cross-range operator. In order to resolve the divergence of Taylor approximation out of the interval of convergence, the rational filter of rectangular type is applied to the depth and cross-range operator. The use of the filter improves the stability of the solution but requires extra numerical burdens. Numerical issues involving the accuracy, efficiency, and stability of the proposed model are discussed and illustrated in an ocean wedge environment.