  Non-linear voter models assume that the opinion of an agent depends on the
opinions of its neighbors in a non-linear manner. This allows for voting rules
different from majority voting. While the linear voter model is known to reach
consensus, non-linear voter models can result in the coexistence of opposite
opinions. Our aim is to derive approximations to correctly predict the time
dependent dynamics, or at least the asymptotic outcome, of such local
interactions. Emphasis is on a probabilistic approach to decompose the opinion
distribution in a second-order neighborhood into lower-order probability
distributions. This is compared with an analytic pair approximation for the
expected value of the global fraction of opinions and a mean-field
approximation. Our reference case are averaged stochastic simulations of a
one-dimensional cellular automaton. We find that the probabilistic second-order
approach captures the dynamics of the reference case very well for different
non-linearities, i.e for both majority and minority voting rules, which only
partly holds for the first-order pair approximation and not at all for the
mean-field approximation. We further discuss the interesting phenomenon of a
correlated coexistence, characterized by the formation of large domains of
opinions that dominate for some time, but slowly change.
