$\beta$-Schemes with Source Terms and the Convergence Analysis

Abstract

The schemes concerned in this study are non-homogeneous $\beta$-schemes for $m = 2$. The homogeneous counterparts (HCPs) of the schemes were constructed by Osher and Chakravarthy (J Oscil Theory Comput Methods Compens Compact 229–274, 1986, [8]). The entire families of $\beta$-schemes are defined for $0<\beta \le (m \left( {\begin{array}{c}2m\\ m\end{array}}\right) )^{(-1)}$, where m is an integer between 2 and 8. These schemes are $2m-1$ order accurate, variation diminishing, $2m+1$ point bandwidth, conservative approximations to the conservation laws. Although the numerical results have been shown to be very effective (Osher and Chakravarthy in J Oscil Theory Comput Methods Compens Compact 229–274, 1986, [8], Osher and Chakravarthy in SIAM J Numer Anal 21:955–984 1984, [7]), the entropy convergence of these schemes has been open. The goal of this paper is to show that, when $m = 2$, $\beta$-schemes with source terms indeed persist entropy consistency for non-homogeneous scalar convex conservation laws by using author’s recent result on extended Yang’s wave tracing theory (Jiang in On wavewise entropy inequalities for high-resolution schemes with source terms II: the fully-discrete case, submitted, [4], Yang in SIAM J Numer Anal 36(1):1–31, 1999, [10]). The entropy convergence of the HCPs of these schemes was established by the author (Jiang in Int J Numer Anal Model 14(1):103–125, 2017, [6]).