We consider the separate convex programming problem with linking linear constraints, where the objective function is in the form of the sum of m individual functions without crossed variables. The special case with m=2 has been well studied in the literature and some algorithms are very influential, e.g. the alternating direction method. The research for the case with general m, however, is in completely infancy, and only a few methods are available in the literature. To generate a new iterate, almost all existing methods produce temporary iterates via solving m subproblems generated by splitting the corresponding augmented Lagrangian function in either parallel or alternating way, and then apply some correction steps to correct the temporary iterates. These correction steps, however, may result in critical difficulties for some applications of the model under consideration. In this paper, we develop the first method requiring no correction steps at all to generate new iterates for the model with general m. We also apply the new method to solve some concrete applications arising in image processing and Statistics, and report the promising numerical performance.