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# CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36 KFUPM Read 25.1-25.4, 26-2, 27-1 CISE301_Topic8L4&5 KFUPM.

## Presentation on theme: "CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36 KFUPM Read 25.1-25.4, 26-2, 27-1 CISE301_Topic8L4&5 KFUPM."— Presentation transcript:

CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36
KFUPM Read , 26-2, 27-1 CISE301_Topic8L4&5 KFUPM

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

Lecture 31 Lesson 4: Runge-Kutta Methods
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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. CISE301_Topic8L4&5 KFUPM

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. CISE301_Topic8L4&5 KFUPM

Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

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. CISE301_Topic8L4&5 KFUPM

Taylor Series in One Variable
Approximation Error CISE301_Topic8L4&5 KFUPM

Taylor Series in One Variable - Another Look -
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Definitions CISE301_Topic8L4&5 KFUPM

Taylor Series Expansion
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Taylor Series in Two Variables
y+k y x x+h CISE301_Topic8L4&5 KFUPM

Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

Runge-Kutta Method Alternative Formula
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Runge-Kutta Method Alternative Formula
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Runge-Kutta Method Alternative Formulas
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Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

Second order Runge-Kutta Method Example
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Second order Runge-Kutta Method Example
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Second order Runge-Kutta Method Example
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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(h3). Fourth order RK have local truncation error of order O(h5). N function evaluations are needed in the Nth order RK method. CISE301_Topic8L4&5 KFUPM

Lecture 32 Lesson 5: Applications of Runge-Kutta Methods to Solve First Order ODEs
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Learning Objectives of Lesson 5
Use Runge-Kutta methods of different orders to solve first order ODEs. CISE301_Topic8L4&5 KFUPM

Runge-Kutta Method CISE301_Topic8L4&5 KFUPM

Runge-Kutta Methods RK2 CISE301_Topic8L4&5 KFUPM

Runge-Kutta Methods RK3 CISE301_Topic8L4&5 KFUPM

Runge-Kutta Methods RK4 CISE301_Topic8L4&5 KFUPM

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. CISE301_Topic8L4&5 KFUPM

Fifth Order Runge-Kutta Methods
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Second Order Runge-Kutta Method
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Second Order Runge-Kutta Method
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Second Order Runge-Kutta Method
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Example 1 Second Order Runge-Kutta Method
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Example 1 Second Order Runge-Kutta Method
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Example 1 Second Order Runge-Kutta Method
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Example 1 Second Order Runge-Kutta Method
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Example 1 Second Order Runge-Kutta Method
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Example 1 Summary of the solution
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Solution after 100 steps CISE301_Topic8L4&5 KFUPM

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

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

Runge-Kutta Methods RK4 CISE301_Topic8L4&5 KFUPM

Example 2 Fourth Order Runge-Kutta Method
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Example 2 Summary of the solution
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Remaining Lessons in Topic 8
Solving Systems of high order ODE Lesson 7: Multi-step methods Lessons 8-9: Methods to solve Boundary Value Problems CISE301_Topic8L4&5 KFUPM

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