SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36

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SE301 Numerical MethodsTopic 8 Ordinary
Differential Equations (ODEs)Lecture 28-36
KFUPM (Term 102) Section 07 Read 25.1-25.4,
26-2, 27-1
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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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Lecture 31Lesson 4 Runge-Kutta Methods
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Learning Objectives of Lesson 4

• To understand the motivation for using Runge
  Kutta method and the basic idea used in deriving
  them.
• To Familiarize with Taylor series for functions
  of two variables.
• Use Runge Kutta of order 2 to solve ODEs.

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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.

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Runge-Kutta Method
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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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Taylor Series in One Variable
Approximation
Error
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Taylor Series in One Variable- Another Look -
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Definitions
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Taylor Series Expansion
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Taylor Series in Two Variables
yk
y
x
xh
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Runge-Kutta Method
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Runge-Kutta Method
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Runge-Kutta Method
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Runge-Kutta Method
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Runge-Kutta MethodAlternative Formula
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Runge-Kutta MethodAlternative Formula
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Runge-Kutta MethodAlternative Formulas
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Runge-Kutta Method
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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

• Runge Kutta 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.

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Lecture 32Lesson 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.

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Runge-Kutta Method
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Runge-Kutta Methods
RK2

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Runge-Kutta Methods
RK3

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Runge-Kutta Methods
RK4

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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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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 1Second Order Runge-Kutta Method

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Example 1Second Order Runge-Kutta Method

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Example 1Second Order Runge-Kutta Method

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Example 1Second Order Runge-Kutta Method

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Example 1Second Order Runge-Kutta Method

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Example 1Summary of the solution

Summary of the solution
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Solution after 100 steps
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Example 24th-Order Runge-Kutta Method
See RK4 Formula

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Example 2Fourth Order Runge-Kutta Method

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Example 2Fourth Order Runge-Kutta Method
See RK4 Formula

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Runge-Kutta Methods
RK4

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Example 2 Fourth Order Runge-Kutta Method

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Example 2Summary of the solution

Summary of the solution
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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