  In this paper, we study the propagation of singularities (in the sense of
$\mathcal{C}^{\infty}$ wave front set) of the solution of a model case
initial-boundary value problem with glancing rays for a concave domain on an
asymptotically anti-de Sitter manifold. The main result addresses the
diffractive problem and establishes that there is no propagation of
singularities into the shadow for the solution, i.e. the diffractive result for
codimension-1 smooth boundary holds in this setting. The approach adopted is
motivated by the work done for a conformally related diffractive model problem
by Friedlander, in which an explicit solution was constructed using the Airy
function. This work was later generalized by Melrose and by Taylor, via the
method of parametrix construction. Our setting is a simple case of
asympotically anti-de Sitter spaces, which are Lorentzian manifolds modeled on
anti-de Sitter space at infinity but whose boundary are not totally geodesic
(unlike the exact anti-de Sitter space). Most technical difficulties of the
problem reduce to studying and constructing a global resolvent for a
semiclassical ODE on $\RR^+$, which at one end is a b-operator (in the sense of
Melrose) while having a scattering behavior at infinity. We use different
techniques near 0 and infinity to analyze the local problems: near infinity we
use local resolvent bounds and near zero we build a local semiclassical
parametrix. After this step, the `gluing' method by Datchev-Vasy serves to
combine these local estimates to get the norm bound for the global resolvent.
