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

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...

Full description

Saved in:
Bibliographic Details
Published in:Applied mathematics and computation 2016-04, Vol.279, p.43-61
Main Authors: Jog, C.S., Agrawal, Manish, Nandy, Arup
Format: Article
Language:English
Subjects:
Chaos theory
Chaotic system
Dynamical systems
Elastodynamics
Finite element method
Mathematical analysis
Mathematical models
Nonlinear dynamics
Nonlinearity
Time finite element method
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2015.12.007