  We consider a Beckmann formulation of an unbalanced optimal transport (UOT)
problem. The $\Gamma$-convergence of this formulation of UOT to the
corresponding optimal transport (OT) problem is established as the balancing
parameter $\alpha$ goes to infinity. The discretization of the problem is
further shown to be asymptotic preserving regarding the same limit, which
ensures that a numerical method can be applied uniformly and the solutions
converge to the one of the OT problem automatically. Particularly, there exists
a critical value, which is independent of the mesh size, such that the discrete
problem reduces to the discrete OT problem for $\alpha$ being larger than this
critical value. The discrete problem is solved by a convergent primal-dual
hybrid algorithm and the iterates for UOT are also shown to converge to that
for OT. Finally, numerical experiments on shape deformation and partial color
transfer are implemented to validate the theoretical convergence and the
proposed numerical algorithm.
