Higher-order Approximations in the Parabolic Equation Method for Water Waves

Application of the parabolic equation method to wave propagation problems implies that the propagation direction of all major components of the wave field must be confined to some narrow band of directions centered on a prechosen principal propagation direction. The allowed directional bandwidth is then delimited by the maximum allowed error in the principal direction wavelength when a transverse wavelength (wave turned at an angle) is imposed. In this paper we investigate higher-order approximations which have the effect of opening the directional bandwidth to better than 45 ø around the principal direction and which fit within the computational framework of the lowerorder approximation. The problem is discussed within the context of the pure-diffraction limit (constant depth), after which a higher-order scheme is provided for waves in domains with slowly varying depth and ambient current. Several computational examples are provided to show the improvements in the higher-order approximation relative to the existing lowerorder approximation. We then examine several forms of computational noise which arise in practical applications of either higheror lower-order approximations and suggest methods for suppressing the nonphysical, computational modes of the solutions.