The split-step Padé solution

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The parabolic equation (PE) method has been the most efficient range-dependent propagation model in ocean acoustics ever since the split-step Fourier solution was developed [F. D. Tappert, ‘‘Numerical solutions of a canonical nonlinear dispersive wave equation,’’ SIAM Rev. 16, 140 (1974)]. Since the asymptotic accuracy of the split-step Fourier solution is restricted to leading order (a serious limitation for some problems), the PE method was extended to higher-order asymptotic accuracy using Padé approximations. Until recently, the higher-order PE has been solved using relatively inefficient finite-difference solutions, and the split-step Fourier solution has remained in widespread use despite the accuracy improvements. To improve efficiency without sacrificing accuracy, we apply Padé approximations to achieve higher-order accuracy in both the numerics and the asymptotics. The split-step Padé solution, which has been implemented for both fluid and elastic media, achieves both higher-order asymptotic accuracy and the efficiency of the split-step Fourier solution. This finite-difference solution is one to three orders of magnitude faster than other finite-difference solutions.