Fast and efficient, high-order accurate methods for the numerical solution of the narrow angle parabolic equation for underwater sound propagation are developed. The space-like derivatives are evaluated using higher-order accuracy. Implicit numerical schemes, which arc second- or higher-order accurate in time-like marching and fourth-order accurate in the space-like direction are presented. The efficiency of various numerical methods for time-like marching is evaluated for Cartesian-type meshes. Furthermore, finite-difference schemes for complex domains are developed where the numerical solution is obtained using irregular meshes and curvilinear coordinate transformations.
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