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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 (Term 101) Section 04 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

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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 (Term 101) Section 04 Read 25.1-25.4, 26-2, 27-1

  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

  3. Lecture 34Lesson 7: Multiple Step Methods

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

  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.

  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

  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

  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

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

  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.

  11. 3-Step Predictor-Corrector

  12. 4-Step Adams-Moulton Predictor-Corrector

  13. How Many Function Evaluations are Done? Number of function evaluations is the Computational Speed or Efficiency How many evaluations per step? No need to repeat the evaluation of function f at previous points Only one new function evaluation in the predictor One function evaluation per correction step # of function evaluations = 1+ number of corrections

  14. Example

  15. Example

  16. Example

  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), y(x-2h)… are used.

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

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