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.