Nonclassical shocks and the Cauchy problem for nonconvex conservation laws

The Riemann problem for a conservation law with a nonconvex (cubic) flux ca n be solved in a class of admissible nonclassical solutions that may violat e the Oleinik entropy condition but satisfy a single entropy inequality and a kinetic relation. We use such a nonclassical Riemann solver in a front t racking algorithm, and prove that the approximate solutions remain bounded in the total variation norm. The nonclassical shocks induce an increase of the total variation and, therefore, the classical measure of total variatio n must be modified accordingly. We prove that the front tracking scheme con verges strongly to a weak solution satisfying the entropy inequality. (C) 1 999 Academic Press.