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.