EE3561_Unit 8Al-Dhaifallah14351 EE 3561 : Computational Methods Unit 8 Solution of Ordinary Differential Equations Lesson 3: Midpoint and Heun’s Predictor corrector Methods

EE3561_Unit 8Al-Dhaifallah14352 Lessons in Topic 8  Lesson 1: Introduction to ODE  Lesson 2: Taylor series methods  Lesson 3: Midpoint and Heun’s method  Lessons 4-5: Runge-Kutta methods  Lesson 6: Solving systems of ODE

EE3561_Unit 8Al-Dhaifallah14353 Learning Objectives of Lesson 3  To be able to solve first order differential equation using Midpoint Method  To be able to solve first order differential equation using Heun’s Predictor Corrector method

EE3561_Unit 8Al-Dhaifallah14354 Outlines of Lesson 3 Lesson 3: Midpoint and Heun’s Predictor-corrector methods Review Euler Method Heun’s Method Midpoint method

EE3561_Unit 8Al-Dhaifallah14355 Euler Method

EE3561_Unit 8Al-Dhaifallah14356 We have seen Taylor series method Euler method is simple but not accurate Higher order Taylor series methods are accurate but require calculating higher order derivatives analytically Introduction

EE3561_Unit 8Al-Dhaifallah14357  The methods proposed in this lesson have the general form  For the case of Euler  Different forms of will be used for the midpoint and Heun’s methods Introduction

EE3561_Unit 8Al-Dhaifallah14358 Midpoint Method

EE3561_Unit 8Al-Dhaifallah14359 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 mid point is calculated This value is used to estimate y i+1.  Local Truncation error of order O(h 3 )  Comparable to Second order Taylor series method

EE3561_Unit 8Al-Dhaifallah Midpoint Method

EE3561_Unit 8Al-Dhaifallah Midpoint Method

EE3561_Unit 8Al-Dhaifallah Midpoint Method

EE3561_Unit 8Al-Dhaifallah Midpoint Method

EE3561_Unit 8Al-Dhaifallah Midpoint Method

EE3561_Unit 8Al-Dhaifallah Midpoint Method

EE3561_Unit 8Al-Dhaifallah Example 1

EE3561_Unit 8Al-Dhaifallah Example 1

EE3561_Unit 8Al-Dhaifallah Summary  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 mid point is calculated This value is used to estimate y i+1.  Local Truncation error of order O(h 3 )  Comparable to Second order Taylor series method

EE3561_Unit 8Al-Dhaifallah Heun’s Predictor Corrector

EE3561_Unit 8Al-Dhaifallah Heun’s Predictor Corrector Method

EE3561_Unit 8Al-Dhaifallah Heun’s Predictor Corrector (Prediction )

EE3561_Unit 8Al-Dhaifallah Heun’s Predictor Corrector (Prediction )

EE3561_Unit 8Al-Dhaifallah Heun’s Predictor Corrector (Prediction )

EE3561_Unit 8Al-Dhaifallah Heun’s Predictor Corrector

EE3561_Unit 8Al-Dhaifallah Heun’s Predictor Corrector

EE3561_Unit 8Al-Dhaifallah Example 2

EE3561_Unit 8Al-Dhaifallah Example 2

EE3561_Unit 8Al-Dhaifallah Summary  Euler, Midpoint and Heun’s methods are similar in the following sense: Different methods use different estimates of the slope  Both Midpoint and Heun’s methods are comparable in accuracy to second order Taylor series method.

EE3561_Unit 8Al-Dhaifallah Comparison Method Local truncation error Global truncation error

EE3561_Unit 8Al-Dhaifallah More in this Unit  Lessons 4-5: Runge-Kutta Methods  Lesson 6: Systems of High order ODE  Lesson 7: Multi-step methods  Lessons 8-9: Boundary Value Problems

EE3561_Unit 8Al-Dhaifallah EE 3561 : Computational Methods Topic 8 Solution of Ordinary Differential Equations Lesson 4: Runge-Kutta Methods

EE3561_Unit 8Al-Dhaifallah Lessons in Topic 8  Lesson 1: Introduction to ODE  Lesson 2: Taylor series methods  Lesson 3: Midpoint and Heun’s method  Lessons 4-5: Runge-Kutta methods  Lesson 6: Solving systems of ODE

EE3561_Unit 8Al-Dhaifallah Learning Objectives of Lesson 4  To understand the motivation for using Runge Kutta method and basic idea used in deriving them.  To Familiarize with Taylor series for functions of two variables  Use Runge Kutta of order 2 to solve ODE

EE3561_Unit 8Al-Dhaifallah Motivation  We seek accurate methods to solve ODE that does not require calculating high order derivatives.  The approach is to suggest a formula involving unknown coefficients then determine these coefficients to match as many terms of the Taylor series expansion

EE3561_Unit 8Al-Dhaifallah Runge-Kutta Method

EE3561_Unit 8Al-Dhaifallah Lecture Taylor Series in Two Variables The Taylor Series discussed in Chapter 4 is extended to the 2-independent variable case. This is used to prove RK formula

EE3561_Unit 8Al-Dhaifallah Taylor Series in One Variable Approximation Error

EE3561_Unit 8Al-Dhaifallah Taylor Series in One Variable another look

EE3561_Unit 8Al-Dhaifallah Definitions

EE3561_Unit 8Al-Dhaifallah Taylor Series Expansion

EE3561_Unit 8Al-Dhaifallah Taylor Series in Two Variables xx+h y y+k

EE3561_Unit 8Al-Dhaifallah Runge-Kutta Method

EE3561_Unit 8Al-Dhaifallah Runge-Kutta Method

EE3561_Unit 8Al-Dhaifallah Runge-Kutta Method

EE3561_Unit 8Al-Dhaifallah Runge-Kutta Method

EE3561_Unit 8Al-Dhaifallah Runge-Kutta Method Alternative Formula

EE3561_Unit 8Al-Dhaifallah Runge-Kutta Method Alternative Formula

EE3561_Unit 8Al-Dhaifallah Runge-Kutta Method Alternative Formulas

EE3561_Unit 8Al-Dhaifallah Runge-Kutta Method

EE3561_Unit 8Al-Dhaifallah Second order Runge-Kutta Method Example

EE3561_Unit 8Al-Dhaifallah Second order Runge-Kutta Method Example

EE3561_Unit 8Al-Dhaifallah Second order Runge-Kutta Method Example

EE3561_Unit 8Al-Dhaifallah143553

EE3561_Unit 8Al-Dhaifallah Summary  Runge Kutta methods generate accurate solution without the need to calculate high order derivatives.  Second order RK have local truncation error of order O(h 3 )  Fourth order RK have local truncation error of order O(h 5 )  N function evaluations are needed in N th order RK method.

EE3561_Unit 8Al-Dhaifallah More in this unit Lesson 5: Applications of Runge-Kutta Methods To solve first order differential equations.  Lessons 6: Solving Systems of high order ODE.