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A class of fully-discrete high-resolution schemes using flux limiters was constructedby P. K. Sweby [SIAM J. Numer. Anal. 21 (1984), 995-1011], which amounted to add a limitedanti-diffusive flux to a first order scheme. This technique has been very successful in obtaininghigh-resolution, second order, oscillation free, explicit difference schemes. However, the entropyconvergence of such schemes has been open. For the scalar convex conservation laws, we use one ofYang's convergence criteria [SIAM. J. Numer. Anal. 36 (1999) No. 1, 1-31] to show the entropyconvergence of the schemes with van Leer's flux limiter when the building block of the schemes isthe Godunov or the Engquish-Osher. The entropy convergence of the corresponding problems insemi-discrete case, for convex conservation laws with or without a source term, has been settled byJiang and Yang [Methods and Applications of Analysis 12 (2005), No. 1, 089-102].

On Convergence of Semi-discrete High Resolution Schemes with van Leer's Flux Limiter for Conservation Laws