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The numerical stability of nonlinear body-wave interaction problems is investigated by applying potential flow assumptions to oscillating, non-wallsided two-dimensional and three-dimensional axisymmetric bodies. This body-wave interaction problem is solved using a mixed two-step Eulerian-Lagrangian method. In the first step, Laplace's equation is solved to determine the unknown potential values on the body and the unknown derivatives of the potentials on the free surface. In the second step, free surface boundary conditions are applied using the results of the first step to find the evolved free surface location and new potential values on the new location. Each step has particular mathematical characteristics (elliptic or parabolic-like), so that each step requires different numerical schemes. Consequently, the numerical stability of this body-wave interaction problem contains the characteristics of both of these two steps. The major contributions made to this body-wave interaction problem are the effects of the various parameters (i.e. time increments, panel length, etc.) and the different forms of the Boundary Integral Method (BIM) on numerical stability and accuracy. The far-field truncation requirement is met by matching the linear outer solution to the nonlinear inner solution at the truncation boundary. The intersection point is traced by the extrapolation method with a special boundary condition at the intersection point. To determine the evolution of the free surface according to a Lagrangian model, a regridding scheme is utilized to prevent the concentration of the Lagrangian markers in the vicinity of high gradients. A parameter for the numerical stability of free surface waves, the Free Surface Stability (FSS) number, is defined as a function of the time step size and the discretized panel length. The various stability regions are investigated by changing the FSS number, Green's function constant c, and numerical schemes. A nonlinear stability analysis is also compared to the results of the linear stability analysis for a number of specific conditions to study the effect of nonlinear boundary conditions.