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

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In the early 1980s, Sweby [19] investigated a class of high resolution schemes using flux limiters for hyperbolic conservation laws. For the convex homogeneous conservation laws, Yang [23] has shown the convergence of the numerical solutions of semi-discrete schemes based on minmod limiter when the general building block of the schemes is an arbitrary $E$-scheme, and based on Chakravarthy-Osher limiter when the building block of the schemes is the Godunov, the Engquist-Osher, or the Lax-Friedrichs to the physically correct solution. Recently, Yang and Jiang [25] have proved the convergence of these schemes for convex conservation laws with a source term. However, the convergence problems of other flux limiter, such as van Leer and superbee have been open. In this paper, we apply the convergence criteria, established in [23] [25] by using Yang's wavewise entropy inequality (WEI) concept, to prove the convergence of the semi-discrete schemes with van Leer's limiter for the aforementioned three building blocks. The result is valid for scalar convex conservation laws in one space dimension with or without a source term. Thus, we have settled one of the aforementioned problems.