We extend the framework and the convergence criteria of wavewise entropy inequalities of [H. Yang, Math. Comp., (1996), pp. 45-67] to a large class of semi-discrete high resolution schemes for hyperbolic conservation laws with source terms. This approach is based on an extended theory of Yang  on wave tracking and wave analysis and the theory of Vol'pert  on BV solutions. For the Cauchy problem of convex conservation laws with source terms, we use one of the criteria to prove the convergence to the entropy solution of generalized MUSCL schemes and a class of schemes using flux limiters previously discussed in 1984 by Sweby.
"ON WAVEWISE ENTROPY INEQUALITIES FOR HIGH-RESOLUTION SCHEMES WITH SOURCE TERMS I: THE SEMI-DISCRETE CASE." Methods Appl. Anal. 10 (4) 487 - 512, Dec 2003.