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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Total variation diminishing Runge-Kutta schemes
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by Sigal Gottlieb and Chi-Wang Shu PDF
Math. Comp. 67 (1998), 73-85 Request permission


In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.
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Additional Information
  • Sigal Gottlieb
  • Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
  • MR Author ID: 358958
  • Email:
  • Chi-Wang Shu
  • Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
  • MR Author ID: 242268
  • Email:
  • Received by editor(s): June 10, 1996
  • Additional Notes: The first author was supported by an ARPA-NDSEG graduate student fellowship.
    Research of the second author was supported by ARO grant DAAH04-94-G-0205, NSF grant DMS-9500814, NASA Langley grant NAG-1-1145 and contract NAS1-19480 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23681-0001, and AFOSR Grant 95-1-0074.
  • © Copyright 1998 American Mathematical Society
  • Journal: Math. Comp. 67 (1998), 73-85
  • MSC (1991): Primary 65M20, 65L06
  • DOI:
  • MathSciNet review: 1443118