The LIP+-stability and error estimates for a relaxation scheme

We show the discrete lip(+)-stability for a relaxation scheme proposed by J in and Xin [Comm. Pure Appl. Math., 48 (1995), pp. 235-277] to approximate convex conservation laws. Equipped with the lip(+)-stability we obtain glob al error estimates in the spaces W-s,W-p for -1 less than or equal to s les s than or equal to 1/p, 1 less than or equal to p less than or equal to inf inity and pointwise error estimates for the approximate solution obtained b y the relaxation scheme. The proof uses the framework introduced by Nessyah u 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.