Accuracy of Lax-Wendroff scheme for discontinuous solutions of convection equations

IT is known from Brenner, Thomee and Wahlbin that the well-known second-order Lax-Wendroff scheme is stable in L~2, but unstable in L~p, p≠2. Generally speaking, if the initialdata is smooth enough and if a difference scheme, which is stable in L~p for some p, has orderof accuracy μ, then we can expect that the solution of the difference scheme converges to thesolution of the differential equation at the rate of order μ in L~p. But for discontinuous solu-tions, which are essential to hyperbolic equations, the above expectation is not true. Error es-timates for discontinuous solutions not only have theoretical meaning, but also practical value.