In the past several years, models for nonlinear wave motions in the basilar membrane based on the hydrodynamic principle and on the biophysical mechanisms of the cochlear function have been developed. The significance of such models can be projected to be their potential applications in designing front-end components of advanced speech processing systems, as well as in serving a powerful tool for studying information processing in central parts of the auditory system. Unfortunately, computer simulations of the models in the past had been exceedingly slow and sometimes unstable, which had prevented use of the models in many possible speech-processing applications. In this paper numerical properties of a finite-difference scheme for the time-domain solution of a one-dimensional nonlinear transmission line basilar membrane model are studied. In particular, the von Neumann method is applied to obtain a sufficient condition under which the finite-difference solution to the model is guaranteed to be stable. The utility of this condition is that it determines the optimal time and spatial mesh sizes, thereby guaranteeing a minimal amount of computation in obtaining stable and accurate model outputs. Also presented are preliminary results on applying the basilar-membrane model, based on the most efficient computation schemes derived from the stability analysis, to process speech signals.