Abstract
Shu and Osher introduced a conservative finite difference discretization for hyperbolic conservation laws using nodal values rather than the traditional cell averages. Their form was obtained by introducing mathematical relations that simplify the resulting numerical methods. Here we instead “derive” their form from the standard cell average approach. In the process, we clarify the origin of their relations and the properties of this formulation. We also investigate the extension of their form to non-uniform grids. We show the strict conservation form only extends to grids with quadratic or exponential stretching. However, a slight generalization can be applied to all smoothly stretched grids with no loss of essential properties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., and Stegun, I. A. (1972). Handbook of Mathematical Functions, Dover Publications, New York, USA, ISBN 0-486-61272-4.
Harten, A. (1995). Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math. 48, 1305-1342.
Harten, A. Multiresolution Representation of Data. II. General Framework, UCLA Computational and Applied Math Report 94-10, April 1994. See also CAM Reports 93-03, 93-06, 93-13, 93-26 (with Donat, R.), 94-12, and 94-21 at www.math.ucla.edu/applied/cam/index.html.
LeVeque, R. J. (1992). Numerical Methods for Conservation Laws, Birkhäuser Verlag, Boston, USA, ISBN 3-8176-2723-5.
Osher, S., and Chakravarthy, S. (1983). Upwind schemes and boundary conditions with applications to euler equations in general geometries. J. Comput. Phys. 50, 447-481.
Shu, C. W., and Osher, S. (1988). Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 429-471.
Shu, C. W., and Osher, S. (1989). Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32-78.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Merriman, B. Understanding the Shu–Osher Conservative Finite Difference Form. Journal of Scientific Computing 19, 309–322 (2003). https://doi.org/10.1023/A:1025312210724
Issue Date:
DOI: https://doi.org/10.1023/A:1025312210724