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Oscillation Theory, Computation, and Methods of Compensated Compactness pp 229–274Cite as

Very High Order Accurate TVD Schemes

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Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 2))

Abstract

A systematic procedure for constructing semi-discrete families of 2m - 1 order accurate, 2m order dissipa-tive, variation diminishing, 2m + 1 point band width, conservation form approximations to scalar conservation laws is presented. Here m is an integer between 2 and 8. Simple first order forward time discretization, used together with any of these approximations to the space derivatives, also results in a fully discrete, variation diminishing algorithm. These schemes all use simple flux limiters, without which each of these fully discrete algorithms is even linearly unstable. Extensions to systems, using a nonlinear field-by-field decomposition are presented, and shown to have many of the same properties as in the scalar case. For linear systems, these nonlinear approximations are variation diminishing, and hence convergent. A new and general criterion for approximations to be variation diminishing is also given. Finally, numerical experiments using some of these algorithms are presented.

Research suppported by NSF Grant No. MCS 82-00788, ARO Grant No. DAAG 29-82 - 0090, NASA Grant No. NAG -1-270, and NASA Consortium Agreement No. NCA 2-1R390-403

Research supported by NASA Grant No. NAG-1–269

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Author information

Authors and Affiliations

  1. Mathematics Department, University of California, Los Angeles, CA, 90024, USA

    Stanley Osher

  2. Rockwell Science Center, 1049 Camino Dos Rios, Thousand Oaks, CA, 91360, USA

    Sukumar Chakravarthy

Authors
  1. Stanley Osher
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  2. Sukumar Chakravarthy
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Editor information

Editors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Constantine Dafermos

  2. School of Mathematics and Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, 55455, USA

    J. L. Ericksen

  3. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    David Kinderlehrer

  4. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Marshall Slemrod

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© 1986 Springer-Verlag New York Inc.

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Osher, S., Chakravarthy, S. (1986). Very High Order Accurate TVD Schemes. In: Dafermos, C., Ericksen, J.L., Kinderlehrer, D., Slemrod, M. (eds) Oscillation Theory, Computation, and Methods of Compensated Compactness. The IMA Volumes in Mathematics and Its Applications, vol 2. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8689-6_9

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  • DOI: https://doi.org/10.1007/978-1-4613-8689-6_9

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8691-9

  • Online ISBN: 978-1-4613-8689-6

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