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CISE301 : Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

CISE301 : 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

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CISE301 : Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

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  1. CISE301: 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 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

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