SPLIT-STEP FINITE-DIFFERENCE AND SPLIT-STEP LANCZOS ALGORITHMS FOR SOLVING ALTERNATIVE HIGHER-ORDER PARABOLIC EQUATIONS

Recently Porter and Jensen [J. Acoust. Soc. Am. 94, 1510-1516 (1993)] reported on a seemingly benign propagation problem in which two wide-a ngle split-step parabolic equations (PEs) performed poorly compared to other PEs. That is, for a source and receiver located within a leaky surface duct, anomalously high transmission losses were predicted for ranges at which the leakage energy is refracted back into the duct. Mo reover, both equations displayed greater sensitivity to the choice of reference wave number than the standard PE. In this paper, a new propa gation operator is formulated that retains the computational efficienc y of the split-step algorithm but is considerably more accurate. Two r apid numerical solution algorithms valid in both two and three dimensi ons are then introduced and applied to the leaky surface duct. To impl ement the simplest second-order method a tridiagonal finite-difference system of equations is solved at each range step in addition to the u sual split-step Fourier computations. Higher-order procedures instead involve applying the Lanczos algorithm solely to the higher-order term s in the wide-angle expansion, thereby circumventing the convergence d ifficulties associated with the direct Lanczos evaluation of the Helmh oltz propagator [Hermansson et al., IEEE J. Light, Technol. 10, 772-77 6 (1992)].