Uniformly High-Order Accurate Nonoscillatory Schemes. I

• Published in: SIAM journal on numerical analysis Vol. 24; no. 2; pp. 279 - 309
• Main Authors: Harten, Ami, Osher, Stanley
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.04.1987
• Subjects: Accuracy; Aeronautics; Applied mathematics; Approximation; Boundary conditions; Cauchy problem; Conservation laws; Consortia; Constant coefficients; Exact sciences and technology; Extrema; Local extrema; Mathematical intervals; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, miscellaneous problems; Research grants; Sciences and techniques of general use; TVD schemes; Conservation law; Approximation; Weak solution; Partial differential equation; Hyperbolic equation; Finite difference method
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/0724022

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.