Nguồn ngoài
Numerically Solving Ordinary Differential Equations
The booklet covers basic Euler solvers, Runge-Kutta, and linear multistep methods and discusses explicit vs. implicit methods. It discusses stability and the stability domains of some of the methods and talks about adaptive methods and the concept of stiffness and concluding with a chapter on symplectic methods and why they are important. Beyond the solver algorithms, the book includes many example ODEs and shows their solutions using the methods discussed. Matlab programs accompany the text.
- Tác giả
- Stuart Brorson
- Nhà xuất bản
- Northeastern University
- Nguồn
- LibreTexts
- Giấy phép
- CC BY-SA
- Ngôn ngữ
- EN
- Danh mục
- Khoa học cơ bản
Mục lục· 60 mục
- 01: Introduction
- 1.01: Solving ordinary differential equations
- 1.02: Sampled functions
- 1.03: Differentiation on the computer
- 1.04: Chapter summary
- 02: Forward Euler method
- 2.01: Forward Euler algorithm
- 2.02: Example - the exponential growth ODE
- 2.03: Local truncation error vs. global solution error
- 2.04: Example - a driven system
- 2.05: Example - the logistic equation
- 2.06: Extending Forward Euler to higher order
- 2.07: Example - the simple harmonic oscillator
- 2.08: Propagator matrix for SHO
- 2.09: Stability
- 2.10: Chapter summary
- 03: Backward Euler method
- 3.01: Backward Euler algorithm
- 3.02: Example - exponential growth ODE
- 3.03: Example - the logistic equation
- 3.04: Example- the simple harmonic oscillator
- 3.05: Chapter summary
- 04: Predictor-corrector methods and Runge-Kutta
- 4.01: Heuns method
- 4.02: Example - the exponential growth ODE and error scaling
- 4.03: Example - the simple harmonic oscillator
- 4.04: Midpoint method
- 4.05: Example - the exponential growth ODE
- 4.06: Runge-Kutta methods
- 4.07: Example - the exponential growth ODE
- 4.08: Example - the simple harmonic oscillator
- 4.09: Example - the hanging pendulum
- 4.10: Example - the Van der Pol oscillator
- 4.11: Example - Duffings equation
- 4.12: Butcher tableaux
- 4.13: Chapter summary
- 05: Adaptive stepping
- 5.01: Choosing h
- 5.02: Example - the logistic equation
- 5.03: Stiff ODEs
- 5.04: Chapter summary
- 06: Linear multistep methods
- 6.01: Trapezoidal method
- 6.02: Example - the exponential growth ODE
- 6.03: Adams-Bashforth method 2nd order
- 6.04: Adams-Bashforth method 3rd order
- 6.05: Adams-Moulton method 3rd order
- 6.06: Example - Adams methods and the Van der Pol oscillator
- 6.07: Generalizations
- 6.08: Chapter summary
- 07: Symplectic integrators
- 7.01: Energy conserving ODEs
- 7.02: Symplectic Euler
- 7.03: Example- the simple harmonic oscillator
- 7.04: Conservative systems phase space and Liouvilles theorem
- 7.05: Euler methods and conservation of phase space volumes
- 7.06: Chapter summary
- 08: Review and summary
- 8.01: Review and summary
- 8.02: Guide to Matlab codes