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Mathematics of Computation

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Total variation diminishing Runge-Kutta schemes


Authors: Sigal Gottlieb and Chi-Wang Shu
Journal: Math. Comp. 67 (1998), 73-85
MSC (1991): Primary 65M20, 65L06
MathSciNet review: 1443118
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Abstract: 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
Email: sg@cfm.brown.edu

Chi-Wang Shu
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: shu@cfm.brown.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-98-00913-2
Keywords: Runge-Kutta method, high order, TVD, low storage
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.
Article copyright: © Copyright 1998 American Mathematical Society