Our contribution in this paper is two fold. First, we propose a novel discretization of the forward model for differential phase-contrast imaging that uses B-spline basis functions. The approach yields a fast and accurate algorithm for implementing the forward model, which is based on the first derivative of the Radon transform. Second, as an alternative to the FBP-like approaches that are currently used in practice, we present an iterative reconstruction algorithm that remains more faithful to the data when the number of projections dwindles. Since the reconstruction is an ill-posed problem, we impose a total-variation (TV) regularization constraint. We propose to solve the reconstruction problem using the alternating direction method of multipliers (ADMM). A specificity of our system is the use of a preconditioner that improves the convergence rate of the linear solver in ADMM. Our experiments on test data suggest that our method can achieve the same quality as the standard direct reconstruction, while using only one-third of the projection data. We also find that the approach is much faster than the standard algorithms (ISTA and FISTA) that are typically used for solving linear inverse problems subject to the TV regularization constraint.