Range-dynamical low-rank split-step Fourier method for the parabolic wave equation

Numerical solutions to the parabolic wave equation are plagued by the curse of dimensionality coupled with the Nyquist criterion. As a remedy, a new range-dynamical low-rank split-step Fourier method is developed. The integration scheme scales sub-linearly with the number of classical degrees of freedom in the transverse directions. It is orders of magnitude faster than the classic full-rank split-step Fourier algorithm and saves copious amounts of storage space. This enables numerical solutions of the parabolic wave equation at higher frequencies and on larger domains, and simulations may be performed on laptops rather than high-performance computing clusters. Using a rank-adaptive scheme to optimize the low-rank equations further ensures the approximate solution is highly accurate and efficient. The methodology and algorithms are demonstrated on realistic high-resolution data-assimilative ocean fields in Massachusetts Bay for two three-dimensional acoustic configurations with different source locations and frequencies. The acoustic pressure, transmission loss, and phase solutions are analyzed in the two geometries with seamounts and canyons across and along Stellwagen Bank. The convergence with the rank of the subspace and the properties of the rank-adaptive scheme are demonstrated, and all results are successfully compared with those of the full-rank method when feasible.