In this paper, an original second-order upwind scheme for convection terms is described and implemented in the context of a Control-Volume Finite-Element Method (CVFEM).The proposed scheme is a second-order extension of the ÿrst-order MAss-Weighted upwind (MAW) scheme proposed by Saabas and Baliga (Numer. Heat Transfer 1994; 26B:381-407). The proposed second-order scheme inherits the well-known stability characteristics of the MAW scheme, but exhibits less artiÿcial viscosity and ensures much higher accuracy. Consequently, and in contrast with nearly all second-order upwind schemes available in the literature, the proposed second-order MAW scheme does not need limiters.Some test cases including two pure convection problems, the driven cavity and steady and unsteady ows over a circular cylinder, have been undertaken successfully to validate the new scheme. The veriÿcation tests show that the proposed scheme exhibits a low level of artiÿcial viscosity in the pure convection problems; exhibits second-order accuracy for the driven cavity; gives accurate reattachment lengths for low-Reynolds steady ow over a circular cylinder; and gives constant-amplitude vortex shedding for the case of high-Reynolds unsteady ow over a circular cylinder.