A class of fully-discrete high-resolution schemes using flux limiters was constructed by P. K. Sweby, which amounted to add a limited anti-diffusive flux to a first order scheme. This technique has been very successful in obtaining high-resolution, second order, oscillation free, explicit difference schemes. However, the entropy convergence of such schemes has been open. For the scalar convex conservation laws, we use one of Yang’s convergence criteria to show the entropy convergence of the schemes with van Leer’s flux limiter when the building block of the schemes is the Godunov or the Engquish-Osher. The entropy convergence of the corresponding problems in semi-discrete case, for convex conservation laws with or without a source term, has been settled by Jiang and Yang.
"On the Convergence of Fully-discrete High-Resolution Schemes with van Leer's Flux Limiter for Conservation Laws." Methods Appl. Anal. 16 (3) 403 - 422, September 2009.