SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36 KFUPM (Term 101) Section 04 Read 25.1-25.4, 26-2, 27-1 CISE301_Topic8L4&5

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

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

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. CISE301_Topic8L4&5

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. CISE301_Topic8L4&5

Second Order Runge-Kutta Method CISE301_Topic8L4&5

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

Taylor Series in One Variable Error Approximation CISE301_Topic8L4&5

Derivation of 2nd Order Runge-Kutta Methods – 1 of 5 CISE301_Topic8L4&5

Derivation of 2nd Order Runge-Kutta Methods – 2 of 5 CISE301_Topic8L4&5

Taylor Series in Two Variables CISE301_Topic8L4&5

Derivation of 2nd Order Runge-Kutta Methods – 3 of 5 CISE301_Topic8L4&5

Derivation of 2nd Order Runge-Kutta Methods – 4 of 5 CISE301_Topic8L4&5

Derivation of 2nd Order Runge-Kutta Methods – 5 of 5 CISE301_Topic8L4&5

2nd Order Runge-Kutta Methods CISE301_Topic8L4&5

Alternative Form CISE301_Topic8L4&5

Choosing , , w1 and w2 CISE301_Topic8L4&5

Choosing , , w1 and w2 CISE301_Topic8L4&5

2nd Order Runge-Kutta Methods Alternative Formulas CISE301_Topic8L4&5

Second order Runge-Kutta Method Example CISE301_Topic8L4&5

Second order Runge-Kutta Method Example CISE301_Topic8L4&5

CISE301_Topic8L4&5

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

2nd Order Runge-Kutta RK2 CISE301_Topic8L4&5

Higher-Order Runge-Kutta Higher order Runge-Kutta methods are available. Derived similar to second-order Runge-Kutta. Higher order methods are more accurate but require more calculations. CISE301_Topic8L4&5

3rd Order Runge-Kutta RK3 CISE301_Topic8L4&5

4th Order Runge-Kutta RK4 CISE301_Topic8L4&5

Higher-Order Runge-Kutta CISE301_Topic8L4&5

Example 4th-Order Runge-Kutta Method RK4 CISE301_Topic8L4&5

Example: RK4 CISE301_Topic8L4&5

4th Order Runge-Kutta RK4 CISE301_Topic8L4&5

Example: RK4 See RK4 Formula Step 1 CISE301_Topic8L4&5

Example: RK4 Step 2 CISE301_Topic8L4&5

Example: RK4 Summary of the solution xi yi 0.0 0.5 0.2 0.8293 0.4 1.2141 CISE301_Topic8L4&5

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) and global truncation error of order O(h2). Higher order RK have better local and global truncation errors. N function evaluations are needed in the Nth order RK method. CISE301_Topic8L4&5

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