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A Remark on the Courant-Friedrichs-Lewy Condition in Finite Difference Approach to PDE’s

  • Kosuke Abe (a1), Nobuyuki Higashimori (a2), Masayoshi Kubo (a3), Hiroshi Fujiwara (a3) and Yuusuke Iso (a3)...
Abstract
Abstract

The Courant-Friedrichs-Lewy condition (The CFL condition) is appeared in the analysis of the finite difference method applied to linear hyperbolic partial differential equations. We give a remark on the CFL condition from a view point of stability, and we give some numerical experiments which show instability of numerical solutions even under the CFL condition. We give a mathematical model for rounding errors in order to explain the instability.

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Corresponding author
Email: k-abe@penta.ge.cst.nihon-u.ac.jp
Email: higashim@econ.hit-u.ac.jp
Email: kubo@i. kyoto-u.ac.jp
Email: fujiwara@acs.i.kyoto-u.ac.jp
Corresponding author. Email: iso@acs.i.kyoto-u.ac.jp
References
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[1]Courant R., Friedrichs K. and Lewy H., Über die partiellen Differenzengleichungen der mathematischen physik, Math. Ann., 100 (1928), pp. 3274.
[2]Courant R., Friedrichs K. and Lewy H., On the partial difference equations of mathematical physics, IBM J. Res. Develop., 11 (1967), pp. 215234 (English translation of the original work [1]).
[3] IEEE Standard for Binary Floating-Point Arithmetic, IEEE Std 754-1985, 1985.
[4]Jezequel F., Round-off error propagation in the solution of the heat equation by finite differences, J. Univ. Comput. Sci., 1 (1995), pp. 469483.
[5]Lax P. D. and Richtmyer R. D., Survey of the stability of linear difference equations, Commun. Pure Appl. Math., 9 (1956), pp. 267293.
[6]Lax P. D., Hyperbolic difference equations: a review of the courant-friedrichs-lewy paper in the light of recent developments, IBM J. Res. Develop., 11 (1967), pp. 235238.
[7]O’Brien G. G., Hyman M. A. and Kaplan S., A study of the numerical solution of partial differential equations, J. Math. Phys., 29 (1951), pp. 223251.
[8]Wilkinson J. H., Rounding Errors in Algebraic Processes, Her Majesty’s Stationery Office, London, 1963.
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Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
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