1 / 18

SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36. KFUPM Read 25.1-25.4, 26-2, 27-1. Outline of Topic 8. Lesson 1: Introduction to ODEs Lesson 2: Taylor series methods Lesson 3: Midpoint and Heun’s method Lessons 4-5: Runge-Kutta methods

aysha
Download Presentation

SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. SE301: Numerical MethodsTopic 8Ordinary Differential Equations (ODEs)Lecture 28-36 KFUPM Read 25.1-25.4, 26-2, 27-1 KFUPM

  2. Outline of Topic 8 • Lesson 1: Introduction to ODEs • Lesson 2: Taylor series methods • Lesson 3: Midpoint and Heun’s method • Lessons 4-5: Runge-Kutta methods • Lesson 6: Solving systems of ODEs • Lesson 7: Multiple step Methods • Lesson 8-9: Boundary value Problems KFUPM

  3. Lecture 34Lesson 7: Multiple Step Methods KFUPM

  4. Outlines of Lesson 7 Solution of ODEs Lesson 7: Adam-Moulton Multi-step Predictor-Corrector Methods KFUPM

  5. Learning Objectives of Lesson 7 • Appreciate the importance of multi-step methods. • Discuss advantages/disadvantages of multi-step methods. • Solve first order ODEs using Adams Moulton multi-step method. KFUPM

  6. Single Step Methods • Single Step Methods: • Euler and Runge-Kutta are single step methods. • Estimates of yi+1 depends only on yi and xi. xi-2 xi-1 xi xi+1 KFUPM

  7. Multi-Step Methods • 2-Step Methods • In a two-step method, estimates of yi+1 depends on yi, yi-1, xi, and xi-1 xi-2 xi-1 xi xi+1 KFUPM

  8. Multi-Step Methods • 3-Step Methods • In an 3-step method, estimates of yi+1 depends on yi ,yi-1 ,yi-2, xi , xi-1, and xi-2 xi-2 xi-1 xi xi+1 KFUPM

  9. Heun’s Predictor Corrector Method Heun’s predictor corrector method is not a multi-step method. KFUPM

  10. 2-Step Predictor-Corrector • At each iteration one prediction step is done • and as many correction steps as needed. • is the estimate of the solution at xi+1 • after k correction steps. KFUPM

  11. 3-Step Predictor-Corrector KFUPM

  12. 4-Step Adams-Moulton Predictor-Corrector KFUPM

  13. How Many Function Evaluations are Done? # of function evaluations = 1+ number of corrections KFUPM

  14. Example KFUPM

  15. Example KFUPM

  16. Example KFUPM

  17. Multi-Step Methods • Single Step Methods • Euler and Runge-Kutta are single step methods. • Information about y(x) is used to estimate y(x+h). • Multistep Methods • Adam-Moulton method is a multi-step method. • To estimate y(x+h), information about y(x), y(x-h), x(x-2h)… are used. KFUPM

  18. Number of Steps • At each iteration, one prediction step is done and as many correction steps as needed. • Usually few corrections steps are done (1 to 3). • It is usually better (in terms of accuracy) to use smaller steps size than corrections beyond few steps. KFUPM

More Related