The Convergence Rate of Finite Difference Schemes in the Presence of Shocks

Finite difference approximations generically have ${\cal O}(1)$ pointwise errors close to a shock. We show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher-order methods. Analytic and numerical examples of this form of accuracy are given. We also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock.