We introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation (and in fact --- more general approximations) to scalar conservation laws with piecewise smooth solutions. To this end, we derive a transport inequality for an appropriately weighted error function. The key ingredient in our approach is a one-sided interpolation inequality between classical $L^1$ error estimates and $Lip^+$ stability bounds. The one-sided interpolation, interesting for its own sake, enables us to convert a global $L^1$ result into a (non-optimal) local estimate. This, in turn, provides the necessary bounds on the coefficients of the above mentioned transport inequality. Estimates on the weighted error then follow from the maximum principal, and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation.
Unlike previous works in this direction, our method can deal with finitely many waves with possible collisions. Moreover, in our approach one does not follow the characteristics but instead makes use of the energy method, and hence this approach could be extended to other types of approximate solutions.
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