Minimax rational approximations for the parabolic equation method

Abstract

Higher‐order Padé approximations have been applied to reduce the limitations and to improve the efficiency of the parabolic equation (PE) method [M. D. Collins, J. Acoust. Soc. Am. 93, 1736–1742 (1993)]. Several of the derivatives are correct at the origin for Padé approximations. Improved efficiency may be achieved by using rational approximations designed using the minimax method, which involves minimizing the maximum error over a domain of interest. This approach has been applied for deriving a PE for fluid media that handles wide propagation angles [Vefring and Mjo/lsnes, J. Acoust. Soc. Am. 93, 1736–1742 (1993)]. Additional applications of the minimax approach include deriving improved approximations for the elastic PE (for which stability is an important issue), the two‐way PE (which may require a careful treatment of the evanescent spectrum), and the split‐step Padé solution (which involves that approximation of a relatively complicated function).