The LIP+ -Stability and Error Estimates for a Relaxation Scheme

We show the discrete lip+-stability for a relaxation scheme proposed by Jin and Xin [Comm. Pure Appl. Math., 48 (1995), pp. 235–277] to approximate convex conservation laws. Equipped with the lip+-stability we obtain global error estimates in the spaces W s,p for −1 ≤ s ≤ 1/p, 1 ≤ p ≤ ∞ and pointwise error estimates for the approximate solution obtained by the relaxation scheme. The proof uses the framework introduced by Nessyahu and Tadmor [SIAM J. Numer. Anal., 29 (1992), pp. 1505–1519]. We also show a maximum principle for the relaxation scheme when the initial data are in an equilibrium state.