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

Jiang, Nan; Yang, Huanan

Methods Appl. Anal. 12, No. 1, 89-101 (2005).

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.

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

Keywords:

conservation law with source terms; schemes with flux limiters; entropy condition convergence; high resolution schemes

Citations:

Zbl 0565.65048; Zbl 0848.65063; Zbl 1077.65099; Zbl 0276.65055