  We study the propagation of singularities in solutions of linear convection
equations with spatially heterogeneous nonlocal interactions. A spatially
varying nonlocal horizon parameter is adopted in the model, which measures the
range of nonlocal interactions. Via heterogeneous localization, this can lead
to the seamless coupling of the local and nonlocal models. We are interested in
understanding the impact on singularity propagation due to the heterogeneities
of nonlocal horizon and the local and nonlocal transition. We first
analytically derive equations to characterize the propagation of different
types of singularities for various forms of nonlocal horizon parameters in the
nonlocal regime. We then use asymptotically compatible schemes to discretize
the equations and carry out numerical simulations to illustrate the propagation
patterns in different scenarios.
