Three-dimensional parabolic-equation solutions with boundary-fitted grids

Abstract

A higher-order operator splitting method incorporating multi-dimensional cross terms has recently been proposed to solve the three-dimensional (3D) parabolic wave equation consisting of a square-root Helmholtz operator. The advantages of this splitting method include providing a more accurate 3D parabolic-equation (PE) approximation, as well as supporting fast marching solvers, such as the Alternating Direction Implicit (ADI) Padé method. To apply this numerical solution scheme to a 3D underwater acoustic waveguide with surface waves in a boundary-fitted model grid, one can employ a one-dimensional (1D) non-uniform discretization formula derived from Galerkin’s method using asymmetric basis functions [W.M. Sanders and M.D. Collins, J. Acoust. Soc. Am. 133, 1953–1958 (2013)]. The use of this discretization formula is to approximate the two alternating sets of 1D differential equations with respect to either one of the two transverse directions at the ADI marching steps. An idealized semi-circular waveguide problem with a closed-form solution is considered as a benchmark to test different 3D PE solutions. Compared to the fixed grid PE solution, the boundary-fitted PE solution is an order of magnitude more accurate in terms of both the magnitude error per unit distance and the relative phase error. [Work supported by the ONR.]