SE 301 Numerical Methods Topic 8 Ordinary Differential

SE 301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28 -36 KFUPM Read 25. 1 -25. 4, 26 -2, 27 -1 CISE 301_Topic 8 L 3 KFUPM 1

Outline of Topic 8 Lesson 1: Introduction to ODEs p Lesson 2: Taylor series methods p Lesson 3: Midpoint and Heun’s method p Lessons 4 -5: Runge-Kutta methods p Lesson 6: Solving systems of ODEs p Lesson 7: Multiple step Methods p Lesson 8 -9: Boundary value Problems p CISE 301_Topic 8 L 3 KFUPM 2

Lecture 30 Lesson 3: Midpoint and Heun’s Predictor Corrector Methods CISE 301_Topic 8 L 3 KFUPM 3

Learning Objectives of Lesson 3 p p 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. CISE 301_Topic 8 L 3 KFUPM 4

Topic 8: Lesson 3: Midpoint and Heun’s Predictor-Corrector Methods • Review Euler Method • Heun’s Method • Midpoint Method CISE 301_Topic 8 L 3 KFUPM 5

Euler Method CISE 301_Topic 8 L 3 KFUPM 6

Introduction p The methods proposed in this lesson have the general form: p For the case of Euler: Different forms of will be used for the Midpoint and Heun’s Methods. p CISE 301_Topic 8 L 3 KFUPM 7

Midpoint Method CISE 301_Topic 8 L 3 KFUPM 8

Motivation p The midpoint can be summarized as: n n n Euler method is used to estimate the solution at the midpoint. The value of the rate function f(x, y) at the mid point is calculated. This value is used to estimate yi+1. Local Truncation error of order O(h 3). p Comparable to Second order Taylor series method. p CISE 301_Topic 8 L 3 KFUPM 9

Midpoint Method CISE 301_Topic 8 L 3 KFUPM 10

Midpoint Method CISE 301_Topic 8 L 3 KFUPM 11

Midpoint Method CISE 301_Topic 8 L 3 KFUPM 12

Midpoint Method CISE 301_Topic 8 L 3 KFUPM 13

Midpoint Method CISE 301_Topic 8 L 3 KFUPM 14

Example 1 CISE 301_Topic 8 L 3 KFUPM 15

Example 1 CISE 301_Topic 8 L 3 KFUPM 16

Heun’s Predictor Corrector CISE 301_Topic 8 L 3 KFUPM 17

Heun’s Predictor Corrector Method CISE 301_Topic 8 L 3 KFUPM 18

Heun’s Predictor Corrector (Prediction) CISE 301_Topic 8 L 3 KFUPM 19

Heun’s Predictor Corrector (Prediction) CISE 301_Topic 8 L 3 KFUPM 20

Heun’s Predictor Corrector (Correction) CISE 301_Topic 8 L 3 KFUPM 21

Example 2 CISE 301_Topic 8 L 3 KFUPM 22

Example 2 CISE 301_Topic 8 L 3 KFUPM 23

Summary p Euler, Midpoint and Heun’s methods are similar in the following sense: n p Different methods use different estimates of the slope. Both Midpoint and Heun’s methods are comparable in accuracy to the second order Taylor series method. CISE 301_Topic 8 L 3 KFUPM 24

Comparison Method CISE 301_Topic 8 L 3 Local truncation error KFUPM Global truncation error 25

More in this Topic Lessons 4 -5: Runge-Kutta Methods p Lesson 6: Systems of High order ODE p Lesson 7: Multi-step methods p Lessons 8 -9: Boundary Value Problems p CISE 301_Topic 8 L 3 KFUPM 26