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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
DOI: https://doi.org/10.1090/S0025-5718-98-00913-2
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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  • 1. M. Carpenter and C. Kennedy, Fourth-order 2N-storage Runge-Kutta schemes, NASA TM 109112, NASA Langley Research Center, June 1994.
  • 2. B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comp., v52, 1989, pp.411-435. MR 90k:65160
  • 3. B. Cockburn, S. Hou and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comp., v54, 1990, pp.545-581. MR 90k:65162
  • 4. A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., v49, 1983, pp.357-393. MR 84g:65115
  • 5. A. Harten, B. Engquist, S. Osher and S. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., v71, 1987, pp.231-303. MR 90a:65199
  • 6. Z. Jackiewicz, R. Renaut and A. Feldstein, Two-step Runge-Kutta methods, SIAM J. Numer. Anal, v28, 1991, pp.1165-1182. MR 92f:65083
  • 7. M. Nakashima, Embedded pseudo-Runge-Kutta methods, SIAM J. Numer. Anal, v28, 1991, pp.1790-1802. MR 92h:65112
  • 8. S. Osher and S. Chakravarthy, High resolution schemes and the entropy condition, SIAM J. Numer. Anal., v21, 1984, pp.955-984. MR 86a:65086
  • 9. A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, New York, 1965. MR 32:8479
  • 10. C.-W. Shu, TVB uniformly high order schemes for conservation laws, Math. Comp., v49, 1987, pp.105-121. MR 89b:65208
  • 11. C.-W. Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., v9, 1988, pp.1073-1084. MR 90a:65196
  • 12. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., v77, 1988, pp.439-471. MR 89g:65113
  • 13. P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., v21, 1984, pp.995-1011. MR 85m:65085
  • 14. B. van Leer, Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method, J. Comput. Phys., v32, 1979, pp.101-136.
  • 15. J.H. Williamson, Low-storage Runge-Kutta schemes, J. Comput. Phys., v35, 1980, pp.48-56. MR 81a:65070

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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: https://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

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