Nonlinear stability of the relaxing schemes for scalar conservation laws Original Research Article

The purpose of this paper is to study nonlinear stability of the relaxing schemes approximating nonconvex scalar conservation laws, constructed by Jin and Xin [4]. We will establish the maximum principle for a first-order and a second-order relaxing schemes presented in [4], if the initial layer is not introduced. Optimal bounds on the total variation and L1-boundedness for the above schemes will also be obtained. Specifically, the conserved physical quantity in the relaxing schemes is TVD. The Lipschitz constant of the L1 continuity in time is shown to be independent of the relaxation parameter ε and the time size k. These imply convergence of the above relaxing schemes.