Methods and Applications of Analysis
Volume 21 (2014)
The convergence of $\alpha$ schemes for conservation laws II: Fully-discrete case
Pages: 201 – 220
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 . The convergence of semi-discrete version of $\alpha$ schemes was proven in a companion paper .
conservation laws, fully-discrete $\alpha$ schemes, entropy convergence
2010 Mathematics Subject Classification
Primary 65M12. Secondary 35L60.