Existing methods for analyzing the stability of a discretized scheme for convection.diffusion terms are usually based on five assumptions, i.e., one-dimensional, linear, first kind of boundary condition, source term free, and uniform grid system. In this article we examine numerically whether deviation from one of the assumptions may enhance the stability of the discretized scheme. The second part of the article is devoted to the criterion of convective boundedness. It is shown that the convective boundedness criterion (CBC) proposed by Gaskell and Lau is only a sufficient condition. Another region in the normalized variable diagram is proposed within which any scheme defined is convectively bounded. Three new bounded high-resolution schemes defined in this region, SBECBC1, 2, and 3, are proposed, and numerical experiments for two advection problems and one diffusion-convection problem demonstrate the high, resolution ability of the SBECBCs for a sharp change in scalar profile.