Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- 17 Initial Value Problems for Ordinary Differential Equations
- 18 Single-Step Methods
- 19 Runge–Kutta Methods
- 20 Linear Multi-step Methods
- 21 Stiff Systems of Ordinary Differential Equations and Linear Stability
- 22 Galerkin Methods for Initial Value Problems
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
21 - Stiff Systems of Ordinary Differential Equations and Linear Stability
from Part IV - Initial Value Problems for Ordinary Differential Equations
Published online by Cambridge University Press: 29 September 2022
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- 17 Initial Value Problems for Ordinary Differential Equations
- 18 Single-Step Methods
- 19 Runge–Kutta Methods
- 20 Linear Multi-step Methods
- 21 Stiff Systems of Ordinary Differential Equations and Linear Stability
- 22 Galerkin Methods for Initial Value Problems
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
Summary
The topic of this chapter is linear stability of schemes for ODEs. The notions of stiffness, linear stability domain, and A-stability are introduced. We discuss the A-stability of Runge-Kutta schemes via the amplification factor. The notion of A-acceptable, Pade approximations of the exponential, and the Hairer-Wanner-Norsett theorem are then presented. This allows us to show that all Gauss-Legendre-Runge-Kutta schemes are A-stable. For multistep schemes we present linear stability criteria and Dahlquist second barrier theorem. The boundary locus method concludes the chapter.
Keywords
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- Chapter
- Information
- Classical Numerical AnalysisA Comprehensive Course, pp. 581 - 595Publisher: Cambridge University PressPrint publication year: 2022