Numerische Mathematik

, Volume 79, Issue 3, pp 397–425

Third order nonoscillatory central scheme for hyperbolic conservation laws

  • Xu-Dong Liu
  • Eitan Tadmor

DOI: 10.1007/s002110050345

Cite this article as:
Liu, X. & Tadmor, E. Numer. Math. (1998) 79: 397. doi:10.1007/s002110050345

Summary.

A third-order accurate Godunov-type scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: 1. A non-oscillatory piecewise-quadratic reconstruction of pointvalues from their given cell averages; and 2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a third-order central scheme, an extension along the lines of the second-order central scheme of Nessyahu and Tadmor \cite{NT}. The scalar scheme is non-oscillatory (and hence – convergent), in the sense that it does not increase the number of initial extrema (– as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann solvers, field-by-field characteristic decompositions, etc., are required. Numerical experiments confirm the high-resolution content of the proposed scheme. Thus, a considerable amount of simplicity and robustness is gained while retaining the expected third-order resolution.

Mathematics Subject Classification (1991):65M10; 65M05

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Xu-Dong Liu
    • 1
  • Eitan Tadmor
    • 2
  1. 1. Department of Mathematics, UCSB, Santa Barbara, CA 93106, USA; e-mail: xliu@math.ucsb.edu US
  2. 2. School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel IL
  3. 3. Department of Mathematics, UCLA, Los Angeles CA 90095, USA; e-mail: tadmor@math.ucla.edu US