Several discrete models for diffusive motion are known to exhibit checkerboard artifacts, absent in their continuous analogues. We study the origins of the checkerboard artifact in the discrete heat equation and show that this artifact decays exponentially in time when following either of two strategies: considering the present state of each lattice site to determine its own future state (self-contributions), or using non-square lattice geometries. Afterwards, we examine the effects of these strategies on nonlinear models of biological cell migration with two kinds of cell-cell interactions: adhesive and polar velocity alignment. In the case of adhesive interaction, we show that growing modes related to pattern formation overshadow artifacts in the long run; nonetheless, artifacts can still be completely prevented following the same strategies as in the discrete heat equation. On the other hand, for polar velocity alignment we show that artifacts are not only strengthened, but also that new artifacts can arise in this model which were not observed in the previous models. We find that the lattice geometry strategy works well to alleviate artifacts. However, the self-contribution strategy must be applied more carefully: lattice sites should contribute to both their own density and velocity values, and their own velocity contribution should be high enough. With this work, we show that these two strategies are effective for preventing artifacts in spatial models based on the discrete continuity equation.