Methods and Applications of Analysis

Volume 21 (2014)

Number 2

The convergence of $\alpha$ schemes for conservation laws II: Fully-discrete case

Pages: 201 – 220

DOI: http://dx.doi.org/10.4310/MAA.2014.v21.n2.a2

Author

Nan Jiang (Department of Mathematical Sciences, University of South Dakota, Vermillion, S.D., U.S.A.)

Abstract

The entropy convergence of a class of fully-discrete $\alpha$ schemes is shown for scalar convex conservation laws in one dimension. These schemes were constructed by S. Osher and S. Chakravarthy in the mid-1980s [1, 12]. When $m = 2$, this class of schemes includes, for different values of $\alpha$, high accuracy (low truncation error) second-order schemes, the conventional second-order accurate upwind total variation diminishing (TVD) scheme and even a third-order accurate TVD scheme. The proof of the entropy consistence is accomplished by using Yang’s wavewise entropy inequality (WEI) framework [17]. The convergence of semi-discrete version of $\alpha$ schemes was proven in a companion paper [6].

Keywords

conservation laws, fully-discrete $\alpha$ schemes, entropy convergence

2010 Mathematics Subject Classification

Primary 65M12. Secondary 35L60.

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