Uniformly High-Order Accurate Nonoscillatory Schemes. I

Abstract

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.

The research of this author was supported by Army Research Office contract DAAG-29-80 C-0041 while he was in residence at the Mathematics Research Center, University of Wisconsin, Madison, Wisconsin 53706, and by National Aeronautics and Space Administration Consortium Agreement NCA2-IR390-403 and Army Research Office grant DAAG 29-82-0090 while at the University of California, Los Angeles, California 90024.

The research of this author was supported by National Science Foundation grant 82-00788, Army Research Office grant DAAG 29-82-0090, National Aeronautics and Space Administration Consortium Agreement NCA2-IR390-403, and National Aeronautics and Space Administration Langley grants NAG 1-270 and NAGI-506.