Advanced Techniques in Numerical Methods for Ordinary Differential Equations
Explore multi-step methods like Adams-Moulton and Heun's predictor-corrector method in solving ODEs. Discover the advantages, disadvantages, and applications of these advanced numerical techniques in differential equations.
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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 1
Outline of Topic 8 Lesson 1: Lesson 2: Lesson 3: Lessons 4-5: Runge-Kutta methods Lesson 6: Solving systems of ODEs Lesson 7: Multiple step Methods Lesson 8-9: Boundary value Problems Introduction to ODEs Taylor series methods Midpoint and Heun s method CISE301_Topic8L7 KFUPM 2
Lecture 34 Lesson 7: Multiple Step Methods CISE301_Topic8L7 KFUPM 3
Outlines of Lesson 7 Solution of ODEs Lesson 7: Adam-Moulton Multi-step Predictor-Corrector Methods CISE301_Topic8L7 KFUPM 4
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 5
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 6
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 7
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 8
Heuns Predictor Corrector Method Heun s predictor corrector method is not a multi-step method. CISE301_Topic8L7 KFUPM 9
2-Step Predictor-Corrector 3 2 1 2 1 2 ( ) ( ) = + 0 i Predictor: y y h f x ,y f x ,y + 1 1 1 i i i i i 1 2 ( ) ( ) = + + 1 k i k i Co rrector: , y y h f x y f x ,y + + + 1 1 1 i i i i At each iteration one prediction step is done and as many correction steps as needed. k iy is the estimate of the solution at xi+1 after k correction steps. + 1 CISE301_Topic8L7 KFUPM 10
3-Step Predictor-Corrector 23 12 5 12 16 12 8 12 5 = + + + Predictor : ( ) f y ( ) ( ) y y h f y f y + 1 -1 -2 i i i i i 12 1 2 = + + + Corrector : ( ) ( ) f y ( ) y y h f y f y + + 1 1 -1 i i i i i 1 CISE301_Topic8L7 KFUPM 11
4-Step Adams Predictor-Corrector (Adams-Bashforth Predictor) Predictor: h y y f y + = + ( ) ) 37 ( + ) 9 ( 55 ( ) 59 ( i ) f y f y f y 1 -1 -2 -3 i i i i i 24 ( Corrector: Adams-Moulto h f y Corrector) n ( ) = + ) 19 ( ) 5 ( f y + + 9 ( ) ( ) y y f y f y + + 1 1 -1 -2 i i i i i i 24 Seepage 774 for ( )and ( ) formulas predictor corr ector CISE301_Topic8L7 KFUPM 12
How Many Function Evaluations are Done? 4-step Adams-Moulton Method h y y + = + Predictor: ( ) ) 37 ( + ) 9 ( 55 ( ) 59 ( i f y ) f y f y f y 1 -1 -2 -3 i i i i i 24 Corrector : h ( ) = + ) 19 ( ) 5 ( f y + + 9 ( f y ) ( ) y y f y f y + + 1 1 -1 -2 i i i i i i 24 How many function evaluations are done per step ? # of function evaluations = 1+ number of corrections CISE301_Topic8L7 KFUPM 13
Example Solve dy dx = + = = 2 2 (0) 2 0.1 x y x y h 2 step predictor c Use orrector method to comput e (0.2) y Weneed 2 step predictor corrector method We will first use two initial con ditio to use the s n RK2 to estim ate (0. 1 ) y CISE301_Topic8L7 KFUPM 14
Example Weneed2initialconditions UseRK2tocompute (0.1)thenwe can use the predictor corrector method dy x y x y dx K K y = + = y = + = = 2 2 (0) 2, 0.1 h = = = 1 2 (0.1) 0.1(0) 0.1(0.2 0.4) 2 0.5(0.06) 0 + = 0.06 2.03 CISE301_Topic8L7 KFUPM 15
Example dy dx = + = = = = = 2 2 (0) 2, (0.1) 2.03, 0.1 x y x y y y y h -1 i i 3 2 1 2 = + Predictor: ( ) f y ( ) y y h f y + 1 -1 i i i i 3 2 1 2 ( ) ( ) ( ) = 2.03 0.1 + 2(0.1) 2.03 0.1 + 0 0 + = 2 2 .1218 1 2 1 2 = + + Corrector: ( ) ( ) f y y y h f y + + 1 1 i i i i 1 2 1 2 ( ) ( ) = 2.03 0.1 + + + = 2 ( ) 2(0.1) 2.03 0.1 2.1256 f y + 1 i CISE301_Topic8L7 KFUPM 16
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), x(x-2h) are used. CISE301_Topic8L7 KFUPM 17
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 18
