Pointwise Error Estimates For Relaxation Approximations to Conservation Laws

Abstract: We obtain sharp {\em pointwise} error estimates for relaxation approximation to scalar conservation laws with piecewise smooth solutions. We first prove that the first partial derivatives for the perturbation solutions are uniformly upper bounded (the so-called $Lip^+$ stability). An one-sided interpolation inequality between classical $L^1$ error estimates and $Lip^+$ stability bounds enables us to convert a global $L^1$ result into a (non-optimal) local estimate. Optimal error bounds on the weighted error then follow from the maximum principle for weakly coupled hyperbolic systems. The main difficulties in obtaining the $Lip^+$ stability and the optimal pointwise errors are how to construct appropriate difference functions'' so that the maximum principle can be applied.

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