Previous work treating shoreline change arising from alongshore sediment transport as a diffusion of shoreline shape has been in error. When the effects of nearshore refraction on breaking-wave characteristics are included, analyses show that shoreline diffusivity varies with deep-water wave characteristics, and that the traditionally used value for shoreline diffusivity is not correct. For sufficiently oblique wave-approach angles, diffusivity becomes negative. As we expand on here, analysis in terms of deep-water wave characteristics shows that a negative diffusivity will occur for common wave-approach angles. Thus, the effects of this instability are likely to be important for many shorelines. Investigating the consequences of this fundamental aspect of shoreline change using a simple numerical model requires important modifications to the 1-line modeling approach, as we detail here. When diffusivity is negative, nonphysical discretization artifacts will appear in such a model unless the algorithm is designed specifically to avoid these artifacts. In addition, the algorithm must be designed to handle model complications related to emergent shoreline structures.