Solution of the second-moment parabolic equation in a refractive medium by the alternating-direction implicit method

The second-moment parabolic equation (PE) simultaneously incorporates refraction, turbulent scattering, and ground interactions. Direct numerical solution of this equation circumvents the need for creating and propagating sound through a large ensemble of randomized turbulence calculations. In previous research, numerical methods were proposed for solving the second-moment PE based on perturbative methods and Cholesky factorization. Here, an efficient computational approach based on the alternating-direction implicit (ADI) method is described. For large ensembles, averages converge on the second-moment ADI calculations. Hence, the ADI calculation can provide a useful benchmark for wave propagation calculations in random media.