KFUPM 1 CISE301 Numerical Methods Topic 8 Ordinary Differential Equations ODEs Lecture 2836 KFUPM Read 251254 262 271 CISE301Topic8L3 KFUPM 2 Outline of Topic 8 Lesson 1 Introduction to ODEs ID: 249726
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CISE301: Numerical MethodsTopic 8 Ordinary Differential Equations (ODEs)Lecture 28-36
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Read 25.1-25.4, 26-2, 27-1Slide2
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Outline of Topic 8Lesson 1: Introduction to ODEsLesson 2: Taylor series methodsLesson 3: Midpoint and Heun’s methodLessons 4-5: Runge-Kutta methodsLesson 6: Solving systems of ODEsLesson 7: Multiple step MethodsLesson 8-9: Boundary value ProblemsSlide3
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Lecture 30Lesson 3: Midpoint and Heun’s Predictor Corrector MethodsSlide4
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Learning Objectives of Lesson 3 To be able to solve first order differential equations using the Midpoint Method.To be able to solve first order differential equations using the Heun’s Predictor Corrector Method.Slide5
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Topic 8: Lesson 3Lesson 3:Midpoint & Heun’s Predictor-Corrector Methods Review Euler Method Midpoint Method Heun’s MethodSlide6
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Euler MethodSlide7
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The methods proposed in this lesson have the general form: For the case of Euler:Different forms of will be used for the Midpoint and Heun’s Methods. IntroductionSlide8
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Midpoint MethodSlide9
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MotivationThe midpoint can be summarized as:Euler method is used to estimate the solution at the midpointThe value of the rate function f(x,y) at the midpoint is calculated and used to estimate yi+1Local Truncation error of order O(h3
)Comparable to 2nd
order
Taylor series
methodSlide10
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Midpoint Method
slopeSlide11
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Midpoint Method
slopeSlide12
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Midpoint Method
slopeSlide13
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Midpoint Method
slopeSlide14
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Midpoint Method
slopeSlide15
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Example 1Slide16
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Example 1Slide17
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Heun’s Predictor CorrectorSlide18
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Heun’s Predictor Corrector Method
Not a power!!
It’s just an indexSlide19
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Heun’s Predictor Corrector(Prediction)Slide20
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Heun’s Predictor Corrector(Prediction)Slide21
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Heun’s Predictor Corrector(Correction)Slide22
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Example 2Slide23
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Example 2Slide24
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SummaryEuler, Midpoint and Heun’s methods are similar in the following sense:Different methods use different estimates of the slope.Both Midpoint and Heun’s methods are comparable in accuracy to the 2nd order Taylor series method.Slide25
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Comparison
Method
Local truncation error
Global truncation errorSlide26
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More in this TopicLessons 4-5: Runge-Kutta MethodsLesson 6: Systems of High order ODELesson 7: Multi-step methodsLessons 8-9: Boundary Value Problems