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
19 - Runge–Kutta Methods
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
We introduce Runge-Kutta methods and their Butcher tableau. We discuss necessary order conditions, and thoroughly analyze some two and three stage schemes. We then discuss the class of Runge-Kutta collocation methods and their consistency. In particular we present the class of Gauss-Legendre-Runge-Kutta methods and their order. Finally, we study how to approximate dissipative equations via so-called dissipative schemes: the M-matrix of a scheme and schemes of B(q) and C(q) types.
Keywords
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- Chapter
- Information
- Classical Numerical AnalysisA Comprehensive Course, pp. 536 - 554Publisher: Cambridge University PressPrint publication year: 2022