  We consider a one-dimensional nonlocal hyperbolic model introduced to
describe the formation and movement of self-organizing collectives of animals
in homogeneous 1D environments. Previous research has shown that this model
exhibits a large number of complex spatial and spatiotemporal aggregation
patterns, as evidenced by numerical simulations and weakly nonlinear analysis.
In this study, we focus on a particular type of localised patterns with
odd/even/no symmetries (which are usually part of snaking solution branches
with different symmetries that form complex bifurcation structures called
snake-and-ladder bifurcations). To numerically investigate the bifurcating
solution branches (to eventually construct the full bifurcating structures), we
first need to understand the numerical issues that could appear when using
different numerical schemes. To this end, in this study, we consider ten
different numerical schemes (the upwind scheme, the MacCormack scheme, the
Fractional-Step method, and the Quasi-Steady Wave-Propagation algorithm,
combining them with high-resolution methods), while paying attention to the
preservation of the solution symmetries with all these schemes. We show several
numerical issues: first, we observe the presence of two distinct types of
numerical solutions (with different symmetries) that exhibit very small errors;
second, in some cases, none of the investigated numerical schemes converge,
posing a challenge for the development of numerical continuation algorithms for
nonlocal hyperbolic systems; lastly, the choice of the numerical schemes, as
well as their corresponding parameters such as time-space steps, exert a
significant influence on the type and symmetry of bifurcating solutions.
