I will present a new technique to deal with the state constraint that binds the transport when source and target have compact support. It takes the form of non-linear boundary conditions which can be combined to a Monge-Ampère equation to solve the optimal transport problem. The wide-stencil discretization technique and fast Newton solver proposed by Oberman and Froese is extended to this framework and allows to compute weak viscosity solution of the optimal transport problem. Numerical solutions will be presented to illustrate strengths and weaknesses of the method.