In recent decades, considerable effort has been devoted to developing higher-order schemes for the discretization of convective terms that are both stable and reliable. In this work, the central idea is that the approximation should be made to reflect the physics of pure convection: the transported quantity is advected along streamlines, and information is propagated only in the upwind direction, i.e., the transported property is determined by previous values along the streamline but not by downstream values. In the proposed approach, streamlines on the computational mesh are represented by discrete streamlines, and the method is called the Discrete Streamline Method (DStreaM). A discrete streamline is constructed as a narrow triangle with one vertex at the node where the approximation is sought and two vertices at upstream neighbouring nodes. Discrete streamlines are oriented according to the local flow direction, in a manner similar to skew-upwind schemes, so that consistency with pure convection is ensured for DStreaM. The method is conservative only for uniform meshes with a constant velocity field; for general meshes and non-uniform velocity fields, it is non-conservative, and a non-zero local conservation error remains. The performance of DStreaM is assessed on the following standard test problems: convection of a step profile, a double-step profile, a sinusoidal profile, and the Smith–Hutton problem. DStreaM solutions are compared with those obtained using the first-order upwind scheme and second-order total variation diminishing (TVD) schemes with Minmod, QUICK, and SUPERBEE limiters. Across these benchmarks, high-resolution solution profiles and L1/L2 error levels comparable to those of the considered TVD schemes are produced by DStreaM. In the DStreaM construction, only local node coordinates and mesh connectivity are used; in this work, implementation is performed on both uniform Cartesian meshes and unstructured triangular meshes generated by a Delaunay triangulation. Representative results are reported with a focus on accuracy, iterative convergence, and conservation limitations.