Contributions to solving initial boundary value problems for partial differential equations have been made by applying finite-difference methods to solve seismic wave propagation problems. Very little has been done in the area of underwater acoustic wave propagation problems, although a set of properly developed numerical methods could very well solve these problems effectively. These numerical methods can solve not only range-dependent problems but also can handle irregular boundaries with arbitrary boundary conditions. In this report, as a start, two accurate general purpose approaches are presented for the solution of variable coefficient parabolic wave equations. In a finite- difference approach, techniques are derived from both the conventional explicit and implicit schemes, and the associated convergence theory is thoroughly analyzed. The techniques are found to be general purpose and to provide reasonable accuracy. In an ordinary differential equation approach the parabolic equation is treated as a system of equations in which the second partial derivative with respect to the space variable is discretized by means of a second order central difference also known as the Method of Lines. Nonlinear multistep NLMS and linear multistep LMS methods are used as predictor-and- corrector for solving this system. A built-in variable step-size technique gives the desired accuracy.