Second order explicit discretization methods for convection dominated transport are studied here from the point of view of discrete minimum and maximum property of numerical solutions. These methods are based on vertex-centered finite volume methods on general unstructured computational grids. It will be shown that “standard TVD methods” [13],[14] do not fulfill in general the discrete minimum and maximum property and that these methods must be modified to obtain numerical solutions with no unphysical oscillations. Finally, new methods based on our theoretical results are proposed.