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Today’s class

Today’s class. Multi-step Methods. Multi-step Methods. R- K methods use only one previous approximation (x i , y i ), known as one-step method. Multi-step methods use several previous points (y i , y i-1 ,…) - explicit (b 0 = 0) & implicit methods.

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Today’s class

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  1. Today’s class • Multi-step Methods Prof. Jinbo Bi CSE, UConn

  2. Multi-step Methods • R- K methods use only one previous approximation (xi, yi), known as one-step method. • Multi-step methods use several previous points (yi, yi-1,…) - explicit (b0 = 0) & implicit methods. • Open & closed formula, non-self start Huen method • Adams-Bashforth Method (explicit methods b0 = 0) • Adams-Moulton Methods (implicit methods) • Predictor-Corrector Methods Prof. Jinbo Bi CSE, UConn

  3. Non-Self-Starting Heun Method • Heun Method • Instead of using an O(h2) forward Euler method as the predictor, use an O(h3) predictor Prof. Jinbo Bi CSE, UConn

  4. Non-Self-Starting Heun Method • Derivation Prof. Jinbo Bi CSE, UConn

  5. Non-Self-Starting Heun Method • Multiple iterations Prof. Jinbo Bi CSE, UConn

  6. Non-Self-Starting Heun Method • Predictor Error • Refine value using predictor modifier • Corrector Error • Refine value using corrector modifier Prof. Jinbo Bi CSE, UConn

  7. Non-Self-Starting Heun Method Prof. Jinbo Bi CSE, UConn

  8. Non-Self-Starting Heun Method • If error compensation is applied Prof. Jinbo Bi CSE, UConn

  9. Predictor-Corrector Methods • Use higher order integration methods as the predictor and corrector • Newton-Cotes • Adams Prof. Jinbo Bi CSE, UConn

  10. Predictor-Corrector Methods • Newton Cotes based • Integrate by fitting an nth degree polynomial to n+1 points Prof. Jinbo Bi CSE, UConn

  11. Predictor-Corrector Methods Newton-Cotes Prof. Jinbo Bi CSE, UConn

  12. Predictor-Corrector Methods • Open Formulas • n=1 • n=2 • n=3 Prof. Jinbo Bi CSE, UConn

  13. Predictor-Corrector Methods • Closed Formulas • n=1 • n=2 Prof. Jinbo Bi CSE, UConn

  14. Adams-Bashforth Methods • Open Formulas • Taylor series expansion • Backward difference substitution for f’ Prof. Jinbo Bi CSE, UConn

  15. Adams-Bashforth Methods • 2nd order • n-th order Prof. Jinbo Bi CSE, UConn

  16. Predictor-Corrector Methods (a) Newton-Cotes (b) Adams Prof. Jinbo Bi CSE, UConn

  17. Adams-Moulton Methods • Closed Formulas • Taylor series expansion • Backward difference substitution for f’ Prof. Jinbo Bi CSE, UConn

  18. Adams-Moulton Methods • 2nd order • n-th order Prof. Jinbo Bi CSE, UConn

  19. Milne’s Method • Predictor-Corrector • Predictor: 3-point Newton-Cotes open • Corrector: 3-point Newton-Cotes closed Prof. Jinbo Bi CSE, UConn

  20. Milne’s Method Prof. Jinbo Bi CSE, UConn

  21. Fourth-Order Adams • Use fourth-order Adams-Bashforth as predictor • Use fourth-order Adams-Moulton as corrector Prof. Jinbo Bi CSE, UConn

  22. Fourth-Order Adams Prof. Jinbo Bi CSE, UConn

  23. Multi-step Methods • Because of the smaller error coefficients, Milne’s Method is slightly more accurate than Fourth-Order Adams methods • However, Milne’s can often be unstable Prof. Jinbo Bi CSE, UConn

  24. Multi-step Methods Prof. Jinbo Bi CSE, UConn

  25. Next class • Ordinary Differential Equations • Boundary-Value Problems • Eigenvalue Problems • Read Chapter 27 • HW 7, Due 11/12 Prof. Jinbo Bi CSE, UConn

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