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
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 CISE301_Topic8L4&5 KFUPM
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 - CISE301_Topic8L4&5 KFUPM
Definitions CISE301_Topic8L4&5 KFUPM
Taylor Series Expansion CISE301_Topic8L4&5 KFUPM
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 CISE301_Topic8L4&5 KFUPM
Runge-Kutta Method Alternative Formula CISE301_Topic8L4&5 KFUPM
Runge-Kutta Method Alternative Formulas CISE301_Topic8L4&5 KFUPM
Runge-Kutta Method CISE301_Topic8L4&5 KFUPM
Second order Runge-Kutta Method Example CISE301_Topic8L4&5 KFUPM
Second order Runge-Kutta Method Example CISE301_Topic8L4&5 KFUPM
Second order Runge-Kutta Method Example CISE301_Topic8L4&5 KFUPM
CISE301_Topic8L4&5 KFUPM
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 CISE301_Topic8L4&5 KFUPM
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 CISE301_Topic8L4&5 KFUPM
Second Order Runge-Kutta Method CISE301_Topic8L4&5 KFUPM
Second Order Runge-Kutta Method CISE301_Topic8L4&5 KFUPM
Second Order Runge-Kutta Method CISE301_Topic8L4&5 KFUPM
Example 1 Second Order Runge-Kutta Method CISE301_Topic8L4&5 KFUPM
Example 1 Second Order Runge-Kutta Method CISE301_Topic8L4&5 KFUPM
Example 1 Second Order Runge-Kutta Method CISE301_Topic8L4&5 KFUPM
Example 1 Second Order Runge-Kutta Method CISE301_Topic8L4&5 KFUPM
Example 1 Second Order Runge-Kutta Method CISE301_Topic8L4&5 KFUPM
Example 1 Summary of the solution CISE301_Topic8L4&5 KFUPM
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 CISE301_Topic8L4&5 KFUPM
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 CISE301_Topic8L4&5 KFUPM
Example 2 Summary of the solution CISE301_Topic8L4&5 KFUPM
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