Pointwise error estimates for relaxation approximations to conservation laws

We obtain sharp pointwise error estimates for relaxation approximation to s calar conservation laws with piecewise smooth solutions. We rst prove that the first-order partial derivatives for the perturbation solutions are unif ormly upper bounded (the so-called Lip(+) stability). A one-sided interpola tion inequality between classical L-1 error estimates and Lip(+) stability bounds enables us to convert a global L-1 result into a ( nonoptimal) local estimate. Optimal error bounds on the weighted error then follow from the maximum principle for weakly coupled hyperbolic systems. The main difficult ies in obtaining the Lip(+) stability and the optimal pointwise errors are how to construct appropriate difference functions so that the maximum princ iple can be applied.