CISE301_Topic8L71 CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture KFUPM (Term 101) Section 04 Read , 26-2, 27-1

CISE301_Topic8L72 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_Topic8L73 L ecture 34 Lesson 7: Multiple Step Methods

CISE301_Topic8L74 Outlines of Lesson 7 Solution of ODEs Lesson 7: Adam-Moulton Multi-step Predictor-Corrector Methods

CISE301_Topic8L75 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_Topic8L76 Single Step Methods  Single Step Methods: Euler and Runge-Kutta are single step methods. Estimates of y i+1 depends only on y i and x i. x i-2 x i-1 x i x i+1

CISE301_Topic8L77 Multi-Step Methods  2-Step Methods In a two-step method, estimates of y i+1 depends on y i, y i-1, x i, and x i-1 x i-2 x i-1 x i x i+1

CISE301_Topic8L78 Multi-Step Methods  3-Step Methods In an 3-step method, estimates of y i+1 depends on y i,y i-1,y i-2, x i, x i-1, and x i-2 x i-2 x i-1 x i x i+1

CISE301_Topic8L79 Heun’s Predictor Corrector Method Heun’s predictor corrector method is not a multi-step method.

CISE301_Topic8L710 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 x i+1 after k correction steps.

CISE301_Topic8L711 3-Step Predictor-Corrector

CISE301_Topic8L712 4-Step Adams-Moulton Predictor-Corrector

CISE301_Topic8L713 How Many Function Evaluations are Done? # of function evaluations = 1+ number of corrections Number of function evaluations is the Computational Speed or Efficiency How many evaluations per step? No need to repeat the evaluation of function f at previous points Only one new function evaluation in the predictor One function evaluation per correction step

CISE301_Topic8L714 Example

CISE301_Topic8L715 Example

CISE301_Topic8L716 Example

CISE301_Topic8L717 Multi-Step Methods  Single Step 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), y(x-2h)… are used.

CISE301_Topic8L718 Number of Steps  At each iteration, one prediction step is done and as many correction steps as needed.  Usually few corrections are done (1 to 3).  It is usually better (in terms of accuracy) to use smaller step size than corrections.