On the Convergence of α Schemes with Source Terms for Scalar Convex Conservation Laws

In this study, we use an extension of Yang’s convergence criterion [N. Jiang, On the wavewise entropy inequality for high-resolution schemes with source terms II: the fully discrete case ] to show the entropy convergence of a class of fully discrete α schemes, now with source terms, for non-homogeneous scalar convex conservation laws in the one-dimensional case. The homogeneous counterparts (HCPs) of these schemes were constructed by S. Osher and S. Chakravarthy in the mid-1980s [ A New Class of High Accuracy TVD Schemes for Hyperbolic Conservation Laws (1985), Very High Order Accurate TVD Schemes (1986)], and the entropy convergence of these methods, when m = 2 , was settled by the author [N. Jiang, The Convergence of α Schemes for Conservation Laws II: Fully-Discrete ]. For semi-discrete α schemes, with or without source terms, the entropy convergence of these schemes was previously established (for m = 2 ) by the author [N. Jiang, The Convergence of α Schemes for Conservation Laws I: Semi-Discrete Case ].