Applied Numerical Mathematics 2 (1986) 347-377 North-Holland 347 zyxwvut SOME RESULTS ON UNIFORMLY HIGH-ORDER ACCURATE ESSENTIALLY NONOSCILLATORY SCHEMES Ami HARTEN * Department of Mathematics, University of California, Los Angeles, CA 90024, U.S.A.; and, Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel Stanley OSHER * and Bjiirn ENGQUIST * Department of Mathematics, University of California, Los Angeles, CA 90024, U.S.A. Sukumar R. CHAKRAVARTHY Rockwell Science Center, Thousand Oaks, CA, U.S.A. We continue the construction and the analysis of essentially 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 an hierarchy of uniformly high-order accurate approximations of any desired order of accuracy which are tailored to be essentially nonoscillatory. This means that, for piecewise smooth solutions, the variation of the numerical approximation is bounded by that of the true solution up to O(hR-’ ), for 0 ( R and h sufficiently small. The design involves an essentially nonoscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approxi- mate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. To solve this reconstruction problem we use a new interpolation technique that when applied to piecewise smooth data gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities. 1. Introduction In this paper we consider numerical approximations to weak solutions of the hyperbolic initial value problem (IVP) u,+f(u), = 0 = u,+ a(u)u,, (1 .la) 24(.X,0) = 240(X). (l.lb) Here u and f are m vectors, and a(u) = af/a u is the Jacobian matrix, which is assumed to have only real eigenvalues and a complete set of linearly independent eigenvectors. The initial data z+,(x) are assumed to be piecewise-smooth functions that are either periodic or of compact support. Let r$ = uh(xi, t,), xj =jh, t, = no, denote a numerical approximation in conservation form. ‘i “+I = U/” - X( [‘+I/* -A-i/z) = (Eh. U’) j. (1.2a) * Research supported by NSF Grant No. DMS85-03294, AR0 Grant No. DAAG29-85-K-0190, NASA Consortuim Agreement No. NCA2-IR390-403, and NASA Langley Grant No. NAGI-270. 0168-9274/86/$3.50 0 1986, Elsevier Science Publishers B.V. (North-Holland)