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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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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