CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36 KFUPM Read 25.1-25.4, 26-2, 27-1 CISE301_Topic8L7 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_Topic8L7 KFUPM

Lecture 34 Lesson 7: Multiple Step Methods CISE301_Topic8L7 KFUPM

Outlines of Lesson 7 Solution of ODEs Lesson 7: Adam-Moulton Multi-step Predictor-Corrector Methods CISE301_Topic8L7 KFUPM

Learning Objectives of Lesson 7 Appreciate the importance of multi-step methods. Discuss advantages/disadvantages of multi-step methods. Solve first order ODEs using Adams Moulton multi-step method. CISE301_Topic8L7 KFUPM

Single Step Methods Single Step Methods: Euler and Runge-Kutta are single step methods. Estimates of yi+1 depends only on yi and xi. xi-2 xi-1 xi xi+1 CISE301_Topic8L7 KFUPM

Multi-Step Methods 2-Step Methods In a two-step method, estimates of yi+1 depend on yi, yi-1, xi, and xi-1 xi-2 xi-1 xi xi+1 CISE301_Topic8L7 KFUPM

Multi-Step Methods 3-Step Methods In an 3-step method, estimates of yi+1 depends on yi ,yi-1 ,yi-2, xi , xi-1, and xi-2 xi-2 xi-1 xi xi+1 CISE301_Topic8L7 KFUPM

Heun’s Predictor Corrector Method Heun’s predictor corrector method is not a multi-step method. CISE301_Topic8L7 KFUPM

2-Step Predictor-Corrector At each iteration one prediction step is done and as many correction steps as needed. is the estimate of the solution at xi+1 after k correction steps. CISE301_Topic8L7 KFUPM

3-Step Predictor-Corrector CISE301_Topic8L7 KFUPM

4-Step Adams Predictor-Corrector CISE301_Topic8L7 KFUPM

How Many Function Evaluations are Done? # of function evaluations = 1+ number of corrections CISE301_Topic8L7 KFUPM

Example CISE301_Topic8L7 KFUPM

Example CISE301_Topic8L7 KFUPM

Example CISE301_Topic8L7 KFUPM

Multi-Step Methods Single Step Methods Multistep Methods Euler and Runge-Kutta are single step methods. Information about y(x) is used to estimate y(x+h). Multistep Methods Adam-Moulton method is a multi-step method. To estimate y(x+h), information about y(x), y(x-h), x(x-2h)… are used. CISE301_Topic8L7 KFUPM

Number of Steps At each iteration, one prediction step is done and as many correction steps as needed. Usually few correction steps are done (1 to 3). It is usually better (in terms of accuracy) to use smaller step sizes than corrections beyond few steps. CISE301_Topic8L7 KFUPM