A new methodology for the Eulerian numerical solution of the advection problem is proposed. The methodology is based on the conservation of both the zero- and the first-order spatial moments inside each element of the computational domain and leads to the solution of several small systems of ordinary differential equations. Since the systems are solved sequentially (one element after the other), the method can be classified as explicit. The proposed methodology has the following properties: (1) it guarantees local and global mass conservation, (2) it is unconditionally stable, and (3) it applies second-order approximation of the concentration and its fluxes inside each element. Limitation of the procedure to irrotational flow fields, for the 2-D and 3-D cases, is discussed. The results of three 1-D and 2-D literature tests are compared with those obtained using other techniques. A new 2-D test, with radially symmetric flow, is also carried out.