1. CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture 28-36
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Read , 26-2, 27-1
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2. 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
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3. Lecture 34 Lesson 7: Multiple Step Methods
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4. Outlines of Lesson 7 Solution of ODEs Lesson 7:
Adam-Moulton Multi-step
Predictor-Corrector Methods
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5. 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.
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6. 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 xi xi xi+1
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7. 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 xi xi xi+1
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8. 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 xi xi xi+1
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9. Heun’s Predictor Corrector Method
Heun’s predictor corrector method is not a multi-step method.
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10. 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.
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11. 3-Step Predictor-Corrector
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12. 4-Step Adams Predictor-Corrector
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13. How Many Function Evaluations are Done?
# of function evaluations = 1+ number of corrections
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14. Example
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15. Example
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16. Example
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17. 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.
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18. 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.
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