  The Lohe sphere model and the Lohe matrix model are prototype continuous
aggregation models on the unit sphere and the unitary group, respectively.
These models have been extensively investigated in recent literature. In this
paper, we propose several discrete counterparts for the continuous Lohe type
aggregation models and study their emergent behaviors using the Lyapunov
function method. For suitable discretization of the Lohe sphere model, we
employ a scheme consisting of two steps. In the first step, we solve the
first-order forward Euler scheme, and in the second step, we project the
intermediate state onto the unit sphere. For this discrete model, we present a
sufficient framework leading to the complete state aggregation in terms of
system parameters and initial data. For the discretization of the Lohe matrix
model, we use the Lie group integrator method, Lie-Trotter splitting method and
Strang splitting method to propose three discrete models. For these models, we
also provide several analytical frameworks leading to complete state
aggregation and asymptotic state-locking.
