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CISE301_Topic8L4&5KFUPM1 CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture KFUPM Read , 26-2, 27-1

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CISE301_Topic8L4&5KFUPM2 Outline of Topic 8 Lesson 1:Introduction to ODEs Lesson 2:Taylor series methods Lesson 3:Midpoint and Heuns method Lessons 4-5: Runge-Kutta methods Lesson 6:Solving systems of ODEs Lesson 7:Multiple step Methods Lesson 8-9:Boundary value Problems

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CISE301_Topic8L4&5KFUPM3 L ecture 31 Lesson 4: Runge-Kutta Methods

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CISE301_Topic8L4&5KFUPM4 Learning Objectives of Lesson 4 To understand the motivation for using Runge-Kutta (RK) method and the basic idea used in deriving them. To get familiar with Taylor series for functions of two variables. To use RK method of order 2 to solve ODEs.

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CISE301_Topic8L4&5KFUPM5 Motivation We seek accurate methods to solve ODEs that do not require calculating high order derivatives. The approach is to use a formula involving unknown coefficients then determine these coefficients to match as many terms of the Taylor series expansion as possible.

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CISE301_Topic8L4&5KFUPM6 Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM7 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.

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CISE301_Topic8L4&5KFUPM8 Taylor Series in One Variable Approximation Error

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CISE301_Topic8L4&5KFUPM9 Taylor Series in One Variable - Another Look -

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CISE301_Topic8L4&5KFUPM10 Definitions

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CISE301_Topic8L4&5KFUPM11 Taylor Series Expansion

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CISE301_Topic8L4&5KFUPM12 Taylor Series in Two Variables xx+h y y+k

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CISE301_Topic8L4&5KFUPM13 Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM14 Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM15 Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM16 Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM17 Runge-Kutta Method Alternative Formula

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CISE301_Topic8L4&5KFUPM18 Runge-Kutta Method Alternative Formula

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CISE301_Topic8L4&5KFUPM19 Runge-Kutta Method Alternative Formulas

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CISE301_Topic8L4&5KFUPM20 Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM21 Second order Runge-Kutta Method Example

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CISE301_Topic8L4&5KFUPM22 Second order Runge-Kutta Method Example

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CISE301_Topic8L4&5KFUPM23 Second order Runge-Kutta Method Example

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CISE301_Topic8L4&5KFUPM24

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CISE301_Topic8L4&5KFUPM25 Summary RK methods generate an 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 the N th order RK method.

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CISE301_Topic8L4&5KFUPM26 L ecture 32 Lesson 5: Applications of Runge-Kutta Methods to Solve First Order ODEs

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CISE301_Topic8L4&5KFUPM27 Learning Objectives of Lesson 5 Use Runge-Kutta methods of different orders to solve first order ODEs.

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CISE301_Topic8L4&5KFUPM28 Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM29 Runge-Kutta Methods RK2

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CISE301_Topic8L4&5KFUPM30 Runge-Kutta Methods RK3

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CISE301_Topic8L4&5KFUPM31 Runge-Kutta Methods RK4

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CISE301_Topic8L4&5KFUPM32 Runge-Kutta Methods Higher order Runge-Kutta methods are available. Higher order methods are more accurate but require more calculations. Fourth order is a good choice. It offers good accuracy with a reasonable calculation effort.

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CISE301_Topic8L4&5KFUPM33 Fifth Order Runge-Kutta Methods

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CISE301_Topic8L4&5KFUPM34 Second Order Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM35 Second Order Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM36 Second Order Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM37 Example 1 Second Order Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM38 Example 1 Second Order Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM39 Example 1 Second Order Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM40 Example 1 Second Order Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM41 Example 1 Second Order Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM42 Example 1 Summary of the solution Summary of the solution

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CISE301_Topic8L4&5KFUPM43 Solution after 100 steps

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CISE301_Topic8L4&5KFUPM44 Example 2 4 th -Order Runge-Kutta Method See RK4 Formula

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CISE301_Topic8L4&5KFUPM45 Example 2 Fourth Order Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM46 Example 2 Fourth Order Runge-Kutta Method See RK4 Formula

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CISE301_Topic8L4&5KFUPM47 Runge-Kutta Methods RK4

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CISE301_Topic8L4&5KFUPM48 Example 2 Fourth Order Runge-Kutta Method

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CISE301_Topic8L4&5KFUPM49 Example 2 Summary of the solution Summary of the solution

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CISE301_Topic8L4&5KFUPM50 Remaining Lessons in Topic 8 Lesson 6: Solving Systems of high order ODE Lesson 7: Multi-step methods Lessons 8-9: Methods to solve Boundary Value Problems

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