Ordinary Differential Equations (ODEs): Midpoint and Heun's Methods Overview

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Explore topics on numerical methods for ODEs, focusing on the Midpoint and Heun's Predictor-Corrector Methods. Understand solving first-order differential equations using these techniques and grasp concepts like local truncation error and global truncation error.

  • ODEs
  • Numerical Methods
  • Midpoint Method
  • Heuns Method
  • Predictor-Corrector
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  1. CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36 KFUPM Read 25.1-25.4, 26-2, 27-1 CISE301_Topic8L3 KFUPM 1

  2. Outline of Topic 8 Lesson 1: Lesson 2: Lesson 3: Lessons 4-5: Runge-Kutta methods Lesson 6: Solving systems of ODEs Lesson 7: Multiple step Methods Lesson 8-9: Boundary value Problems Introduction to ODEs Taylor series methods Midpoint and Heun s method CISE301_Topic8L3 KFUPM 2

  3. Lecture 30 Lesson 3: Midpoint and Heun s Predictor Corrector Methods CISE301_Topic8L3 KFUPM 3

  4. Learning Objectives of Lesson 3 To be able to solve first order differential equations using the Midpoint Method. To be able to solve first order differential equations using the Heun s Predictor Corrector Method. CISE301_Topic8L3 KFUPM 4

  5. Topic 8: Lesson 3 Lesson 3: Midpoint & Heun s Predictor-Corrector Methods Review Euler Method Midpoint Method Heun s Method CISE301_Topic8L3 KFUPM 5

  6. Euler Method Problem ( ) ( y x Euler Method ( y y x y y for i = = ( , ) f x y y ) y x 0 0 = = + ) ( , ) h f x y = + 0 0 1 i i i i 1,2,... 2 Local Truncation Error Global Truncation Error ) O(h O(h ) CISE301_Topic8L3 KFUPM 6

  7. Introduction to Problem f x y = solved be y y is x a first y = order ODE : ( ) ( , ), ( 0) x 0 The methods proposed in this lesson have the general form: h y y i i + = +1 = ( , ) f iy x For the case of Euler: Different forms of will be used for the Midpoint and Heun s Methods. i CISE301_Topic8L3 KFUPM 7

  8. Midpoint Method Problem ( ) y x Midpoint Method ( ) y x h y f x y + = = ( , ) f x y y 0 0 = = ( ) ( , ) y x y y 0 0 1 2 i i i 2 + i = + ( , ) y y h f x y + 1 1 2 1 2 i i + + i i 3 Local Truncation Error Global Truncation Error ) ) O(h 2 O(h CISE301_Topic8L3 KFUPM 8

  9. Motivation The midpoint can be summarized as: Euler method is used to estimate the solution at the midpoint The value of the rate function f(x,y) at the midpoint is calculated and used to estimate yi+1 Local Truncation error of order O(h3) Comparable to 2nd order Taylor series method CISE301_Topic8L3 KFUPM 9

  10. Midpoint Method ( , ) iy x i x x x + 0 1 1 i + i 2 h = + = + ( , , ) i ( , ) y y f x y y y h f x y + 1 1 1 1 i i i i 2 + + + i i i 2 2 2 slope CISE301_Topic8L3 KFUPM 10

  11. Midpoint Method slope = ( , ) f iy x i ( , ) iy x i x x x + 0 1 1 i + i 2 h = + = + ( , , ) i ( , ) y y f x y y y h f x y + 1 1 1 1 i i i i 2 + + + i i i 2 2 2 slope CISE301_Topic8L3 KFUPM 11

  12. Midpoint Method ( , ) x y 1 1 slope = ( , ) f iy x + + i i 2 2 i ( , ) iy x i x x x + 0 1 1 i + i 2 h = + = + ( , , ) i ( , ) y y f x y y y h f x y + 1 1 1 1 i i i i 2 + + + i i i 2 2 2 slope CISE301_Topic8L3 KFUPM 12

  13. Midpoint Method = ( , ) slope f x y 1 1 + + i i ( , ) x y 2 2 1 1 + + i i 2 2 ( , ) iy x i x x x + 0 1 1 i + i 2 h = + = + ( , , ) i ( , ) y y f x y y y h f x y + 1 1 1 1 i i i i 2 + + + i i i 2 2 2 slope CISE301_Topic8L3 KFUPM 13

  14. Midpoint Method = ( , ) slope f x y 1 1 + + i i ( , ) x y 2 2 1 1 + + i i 2 2 ( , ) iy x i x x x + 0 1 1 i + i 2 h = + = + ( , , ) i ( , ) y y f x y y y h f x y + 1 1 1 1 i i i i 2 + + + i i i 2 2 2 slope CISE301_Topic8L3 KFUPM 14

  15. Example 1 Midpoint the Use + = x y Method solve to the ODE + 2 ( ) 1 x y = h ) 0 ( y 1 = Use Determine . 1 . 0 y(0.1) and y(0.2) CISE301_Topic8L3 KFUPM 15

  16. Example 1 = + + = = = = 2 Problem: ( , ) 1 , ( ) (0) 1, 0.1 f x y y x y y x y h 0 0 = Step1 ( 0 ) h : i = + ) 1 = + 0.05(1 0 1) + + = ( , f x y 1.1 y y 1 2 = 0 0 0 2 + 0 + 1 0.1(1 = + + + = ( , ) 1.1 0.0025) 1.2103 y y h f x y 1 0 1 2 1 2 + + 0 0 = Ste p2 ( 1) h : i = + = + + + = ( , f x y ) 1. 2103 .05(1 1.2 1 3 0 0.01) 1.3213 y y 1 2 = 1 1 1 2 + 1 + = + + + = ( , ) 1.21 03 0.1(1 1.32 13 0.0225) 1.4446 8 y y h f x y 2 1 1 2 1 2 + + 1 1 CISE301_Topic8L3 KFUPM 16

  17. Heuns Predictor Corrector CISE301_Topic8L3 KFUPM 17

  18. Heuns Predictor Corrector Method Problem ( ) y x H ' Method ( y x = = + eun s y = ( , ) f x y ) 0 0 = 0 i ( ) Predictor: ( , ) y x y y y h f x y h + + 0 0 1 i i i ( ) + = + 1 k i k i Corrector: ( , f x y ) ( , ) y y f x y + + + 1 1 1 i i i i 2 3 Local Truncation Error Global Truncation Error ) ) O(h Not a power!! It s just an index 2 O(h CISE301_Topic8L3 KFUPM 18

  19. Heuns Predictor Corrector (Prediction) 0 + i ( , ) x y + 1 1 i ( , ) iy x i x x + 1 i i 0 i = + Prediction ( , ) y y h f x y + 1 i i i CISE301_Topic8L3 KFUPM 19

  20. Heuns Predictor Corrector (Prediction) 0 + i ( , ) x y + = 0 + i 1 1 i ( , ) slope f x y + 1 1 i ( , ) iy x i x x + 1 i i = + 0 i Prediction ( , ) y y h f x y + 1 i i i CISE301_Topic8L3 KFUPM 20

  21. Heuns Predictor Corrector (Correction) + 0 + i ( , ) ( , ) f x y f x y = + 1 1 i i i slope 2 0 + i ( , ) x y + 1 1 i 1 i ( , ) x y ( , ) iy x + + 1 1 i i x x + 1 i i ( ) ) 1 + h = + + 1 i 0 i ( , ) ( , y y f x y f x y + + 1 1 i i i i 2 CISE301_Topic8L3 KFUPM 21

  22. Example 2 Use the Heun's Method to solve the ODE ( ) 1 (0) 1 Us one correction onl e 0.1, and Determine y(0.1 ) and y(0.2) = + = = + 2 y x y x y y h CISE301_Topic8L3 KFUPM 22

  23. Example 2 = + + = = = = 2 Problem: ( , ) 1 , ( y x ) (0) y 1, 0.1 f x y y x y h 0 0 Step1 ( = 0): i = + ) 1 0.1(2) = + = 0 1 Predictor: ( , 1.2 y y h f x y h 0 0 0 ( ) ) = + + 1 1 0 1 Corrector: ( , f x y ) ( , f x y ) y y 0 0 0 1 2 ( 1 0.05 = + + 1 0.05(2 = + + = St ep2 ( = (0,1 ) (0.1, 1 .2 ) 2.21) 1. 2105 f f 1): i = + = + = 0 2 Predictor: ( , ) 1 .2105 0.1 (0.1, f 1.2105 ) 1.4326 y y h f x y h + 1 1 1 ( ) = + 1 2 0 2 Corrector: ( , f x y ) ( , f x ) y y y 1 1 1 2 2 = + + = 1.2105 0.05( (0.1, 1.210 5 ) ( 0.2, 1.4326 )) 1.44 5 2 f f CISE301_Topic8L3 KFUPM 23

  24. Summary Euler, Midpoint and Heun s methods are similar in the following sense: slope h y y i i + = +1 Different methods use different estimates of the slope. Both Midpoint and Heun s methods are comparable in accuracy to the 2nd order Taylor series method. CISE301_Topic8L3 KFUPM 24

  25. Comparison Method Local truncation error Global truncation error = + 2 Euler Method ( , ) ( ) ( ) y y h f x y O h O h + 1 i i i i Heun' Method s = + 0 + i 3 2 Predictor : ( , ) ( ) ( ) y y h f x y O h O h 1 i i i ( ) h + = + + 1 k i k i Corrector : ( , ) ( , ) y y f x y f x y + + + 1 1 1 i i i i 2 h = + 3 2 Midpoint ( , ) ( ) ( ) y y f x y O h O h 1 i i i 2 + i 2 = + ( , ) y y h f x y + 1 1 1 i i + + i i 2 2 CISE301_Topic8L3 KFUPM 25

  26. More in this Topic Lessons 4-5: Runge-Kutta Methods Lesson 6: Systems of High order ODE Lesson 7: Multi-step methods Lessons 8-9: Boundary Value Problems CISE301_Topic8L3 KFUPM 26

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