Approximate solutions to slightly viscous conservation laws

We study approximate solutions of a slightly viscous conservation law in one dimension, constructed by two asymptotic expansions that are cut off after the third order terms. In the shock layer an inner solution is valid and an outer solution is valid elsewhere. The two solutions are matched in a matching region. Based on the stability results in [10] we show that for a given time interval the difference between the approximate solutions and the true solution is not larger than O(e), where e is the viscosity coefficient.