A finite difference method is developed to solve the three-dimensional, steady, incompressible, potential flow equations obtained by using a potential function, o, and two mutually orthogonal stream functions, u and u*, to describe the flow. Problems are formulated in an inverse space where the potential function and the two stream functions are the independent variables, and the Cartesian coordinates x, y, and z are the dependent variables. The boundaries of the problem in the physical space, including the free surface, have known positions in the inverse space, so trial and error adjustments to the positions of the boundaries are unnecessary. Methods of describing the effect of the placement of a body, whose shape is partially specified, in the flow field are developed using finite differences, and a solution for the x-, y-, and z-coordinates is obtained at each grid point formed by the intersection of surfaces held constant with respct to o, u, and u* in the inverse space.
Davis, Allen L. and Jeppson, Roland W., "Solving Three-Dimensional Potential Flow Problems by Means of an Inverse Formulation and Finite Differences" (1973). Reports. Paper 517.