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The time finite element as a robust general scheme for solving nonlinear dynamic equations including chaotic systems

Author

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  • Jog, C.S.
  • Agrawal, Manish
  • Nandy, Arup

Abstract

Schemes that can be proven to be unconditionally stable in the linear context can yield unstable solutions when used to solve nonlinear dynamical problems. Hence, the formulation of numerical strategies for nonlinear dynamical problems can be particularly challenging. In this work, we show that time finite element methods because of their inherent energy-momentum conserving property (in the case of linear and nonlinear elastodynamics), provide a robust time-stepping method for nonlinear dynamic equations (including chaotic systems). We also show that most of the existing schemes that are known to be robust for parabolic or hyperbolic problems can be derived within the time finite element framework; thus, the time finite element provides a unification of time-stepping schemes used in diverse disciplines. We demonstrate the robust performance of the time finite element method on several challenging examples from the literature where the solution behavior is known to be chaotic.

Suggested Citation

  • Jog, C.S. & Agrawal, Manish & Nandy, Arup, 2016. "The time finite element as a robust general scheme for solving nonlinear dynamic equations including chaotic systems," Applied Mathematics and Computation, Elsevier, vol. 279(C), pages 43-61.
    • Handle: RePEc:eee:apmaco:v:279:y:2016:i:c:p:43-61
    DOI: 10.1016/j.amc.2015.12.007
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    Cited by:

    1. Xueqing Shi & Daniel (Jian) Sun & Ying Zhang & Jing Xiong & Zhonghua Zhao, 2020. "Modeling Emission Flow Pattern of a Single Cruising Vehicle on Urban Streets with CFD Simulation and Wind Tunnel Validation," IJERPH, MDPI, vol. 17(12), pages 1-17, June.
    2. Agrawal, Manish & Jog, C.S., 2017. "A quadratic time finite element method for nonlinear elastodynamics within the context of hybrid finite elements," Applied Mathematics and Computation, Elsevier, vol. 305(C), pages 203-220.

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