##
**On convergence of semi-discrete high resolution schemes with van Leer’s flux limiter for conservation laws.**
*(English)*
Zbl 1128.65071

Summary: In the early 1980s, P. K. Sweby [SIAM J. Numer. Anal. 21, 995–1011 (1984; Zbl 0565.65048)] investigated a class of high resolution schemes using flux limiters for hyperbolic conservation laws. For the convex homogeneous conservation laws, H. Yang [Math. Comput. 65, No. 213, 45–67; Supplement S1–S13 (1996; Zbl 0848.65063)] has shown the convergence of the numerical solutions of semi-discrete schemes based on Chakravarthy-Osher 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 one to the physically correct solution. Recently, H. Yang and N. Jiang [Methods Appl. Anal. 10, No. 4, 487–512 (2003; Zbl 1077.65099)] have proved the convergence of these schemes for convex conservation laws. However, the convergence problems of other flux limiter, such as B. van Leer’s [J. Comput. Phys. 14, 361–370 (1974; Zbl 0276.65055)] and superbee have been open.

In this paper, we apply the convergence criteria, established 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.

In this paper, we apply the convergence criteria, established 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.

### MSC:

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |