Michael D. Collins
Naval Res. Lab., Washington, DC 20375
Higher-order Pade 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 Pade 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 Mj(slashed oh)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 Pade solution (which involves that approximation of a relatively complicated function).