UNIFORMLY HIGH-ORDER ACCURATE NONOSCILLATORY SCHEMES .1.

We begin the construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper we construct a uniformly second-order approximation, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise-linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem and an average of this approximate solution over each cell.