The elastic behavior of the sucker-rod string, in most models, is simulated by a second-order wave equation with a viscous damping factor, which does not consider the inertia effects of the fluid surrounding the rod string. To consider these effects, another system of first-order wave equations must be solved simultaneously with the rod string equation. Solution of the system of partial differential equation becomes even more complicated if the sucker-rod string properties are not uniform. Because tapered and combination rods are often used in the string, a new approach has been developed. This approach can simulate any combination of rods and fluid inertia effects in a relatively simple and effective manner. It applies the Bergeron method, which can be used for pressure and stress propagation in fluids and solids respectively, by using a system of linear algebraic equations instead of the wave equation(s). A simple example problem is solved before applying the approach to a sucker-rod pumping system. The numerical results are compared to the actual measured surface dynamometer cards. Analysis of the numerical results shows that the fluid inertia effects are important for accurate modeling and design of the sucker-rod pumping system.