To realize the implementation of TVD schemes on unstructured grids, an improved r-factor algorithm is proposed for the numerical simulation of the convection-dominated problems. The distance from the face center to the auxiliary upwind node is considered in the new r-factor algorithm, which achieves more reasonable prediction of the far-upwind node position. The accuracy of TVD schemes on unstructured grids is improved by invoking an interpolation scheme that contains more upwind information and considering the physical deviation between the actual nodes and the auxiliary nodes. Meanwhile, the computational efficiency is improved by avoiding the introduction of complex vector computation procedures in the new algorithm. Benchmark testing results of the convection-dominated problems on unstructured grids indicate that the new r-factor algorithm outperforms existing ones in terms of both accuracy and efficiency.