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A study of spurious asymptotic numerical solutions of nonlinear differential equations by the nonlinear dynamics approach

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Twelfth International Conference on Numerical Methods in Fluid Dynamics

Part of the book series: Lecture Notes in Physics ((LNP,volume 371))

Abstract

The goal of this paper is to present some new results and to give insight and guidelines on the application of nonlinear dynamics theory to the better understanding of asymptotic numerical solutions and nonlinear instability in finite difference methods for nonlinear differential equations that display genuinely nonlinear behavior. Our hope is to reach researchers in the fields of computational sciences and, in particular, computational fluid dynamics (CFD). Although the study of nonlinear dynamics and chaotic dynamics for nonlinear differential equations and for nonlinear discrete maps (nonlinear difference equations) have independently flourished for the last decade, there are very few investigators addressing the issue on the connection between the nonlinear dynamical behavior of the continuous systems and the corresponding nonlinear discrete map resulting from finite-difference discretizations. This issue is especially vital for computational sciences since nonlinear differential equations in applied sciences can rarely be solved in closed form and it is often necessary to replace them by finite dimensional nonlinear discrete maps. It is important to realize that these nonlinear discrete maps can exhibit a much richer range of dynamical behavior than their continuum counterparts. Furthermore, it is important to ask what happens when linear stability in numerical integrations breaks down for problems with genuinely nonlinear behavior. Here our objective is neither to provide theory nor to illustrate with realistic examples the connection of the dynamical behavior of practical fluid dynamics equations with their discretized counterparts, but rather to give insight into the nonlinear features unconventional to this type of study and to concentrate on the fundamental ideas.

Staff Scientist, Fluid Dynamics Division.

Lecturer, Department of Mathematics.

Senior Lecturer, Department of Mathematical Sciences.

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Author information

Authors and Affiliations

  1. NASA Ames Research Center, 94035, Moffett Field, CA, USA

    H. C. Yee

  2. University of Reading, RG6 2AX, Whiteknights, Reading, England

    P. K. Sweby

  3. University of Dundee, DD1 4HN, Dundee, Scotland

    D. F. Griffiths

Authors
  1. H. C. Yee
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  2. P. K. Sweby
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  3. D. F. Griffiths
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K. W. Morton

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© 1990 Springer-Verlag

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Yee, H.C., Sweby, P.K., Griffiths, D.F. (1990). A study of spurious asymptotic numerical solutions of nonlinear differential equations by the nonlinear dynamics approach. In: Morton, K.W. (eds) Twelfth International Conference on Numerical Methods in Fluid Dynamics. Lecture Notes in Physics, vol 371. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-53619-1_180

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  • DOI: https://doi.org/10.1007/3-540-53619-1_180

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53619-2

  • Online ISBN: 978-3-540-46918-6

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