### ASA 125th Meeting Ottawa 1993 May

## 2aAO2. The split-step Pade solution.

**Michael D. Collins
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*Naval Res. Lab., Washington, DC 20375
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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 Pade 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 Pade approximations to achieve
higher-order accuracy in both the numerics and the asymptotics. The split-step
Pade 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.