What question did this study set out to answer?

Analyze the global asymptotic nonlinear behavior of iterative implicit linear multistep methods when solving systems of nonlinear ordinary differential equations.

December 1, 1994International Journal of Bifurcation and ChaosOpen Access

Global Asymptotic Behavior of Iterative Implicit Schemes

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Authors

HYH. C. Yee PSP. K. Sweby

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Overview

Numerical analysis demonstrates stability changes and basin distortions in iterative implicit multistep schemes, indicating mechanisms of nonconvergence in computational fluid dynamics.

Key Points

Analyze the global asymptotic nonlinear behavior of iterative implicit linear multistep methods when solving systems of nonlinear ordinary differential equations.
Evaluated four implicit linear multistep methods applied to three autonomous 2×2 nonlinear ordinary differential equation systems using dynamical systems theory.
Tested simple iteration as well as full and modified Newton iterations within the discretization schemes.
Benchmarked performance against explicit Runge-Kutta methods, a noniterative implicit procedure, and Newton's method for steady-state solutions.
All four implicit linear multistep methods generated spurious asymptotes, altered steady-state stability, and severely distorted true basins of attraction.
The noniterative implicit procedure mirrored natural basins of attraction more accurately and operated with higher efficiency than iterative implicit schemes.
Newton's method converged to the exact steady state even with initial data far from the solution, disproving the necessity of near-solution initializations.

Cite This Study

Yee et al. (1994) studied this question.

synapsesocial.com/papers/6a71c92ba2d7cf2e39c2db1c https://doi.org/10.1142/s0218127494001210

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December 1, 1994International Journal of Bifurcation and ChaosOpen Access

Global Asymptotic Behavior of Iterative Implicit Schemes

View Full Paper

Ask AI

Bookmark

Authors

HYH. C. Yee PSP. K. Sweby

Discussion

Member takes

Overview

Numerical analysis demonstrates stability changes and basin distortions in iterative implicit multistep schemes, indicating mechanisms of nonconvergence in computational fluid dynamics.

Key Points

Analyze the global asymptotic nonlinear behavior of iterative implicit linear multistep methods when solving systems of nonlinear ordinary differential equations.
Evaluated four implicit linear multistep methods applied to three autonomous 2×2 nonlinear ordinary differential equation systems using dynamical systems theory.
Tested simple iteration as well as full and modified Newton iterations within the discretization schemes.
Benchmarked performance against explicit Runge-Kutta methods, a noniterative implicit procedure, and Newton's method for steady-state solutions.
All four implicit linear multistep methods generated spurious asymptotes, altered steady-state stability, and severely distorted true basins of attraction.
The noniterative implicit procedure mirrored natural basins of attraction more accurately and operated with higher efficiency than iterative implicit schemes.
Newton's method converged to the exact steady state even with initial data far from the solution, disproving the necessity of near-solution initializations.

Cite This Study

Yee et al. (1994) studied this question.

synapsesocial.com/papers/6a71c92ba2d7cf2e39c2db1c https://doi.org/10.1142/s0218127494001210

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