Methods and Applications of Analysis
Volume 12 (2005)
On Convergence of Semi-discrete High Resolution Schemes with van Leer's Flux Limiter for Conservation Laws
Pages: 89 – 102
In the early 1980s, Sweby  investigated a class of high resolution schemes using flux limiters for hyperbolic conservation laws. For the convex homogeneous conservation laws, Yang  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  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   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.