The Computable Differential Equation Lecture ... - Bruce E. Shapiro

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80 CHAPTER 4. IMPROVING ON EULER’S METHOD Figure 4.5: Result of the forward Euler method to solve y ′ = −100(y−sin t), y(0) = 1 with h = 0.001 (top), h = 0.019 (middle), and h = 0.02 (third). The bottom figure shows the same equation solved with the backward Euler method for step sizes of h = 0.001, 0.02, 0.1, 0.3, left to right curves, respectively . method: For the scalar test equation this gives y n = y n−1 + h n f(t n , y n ) (4.98) y n = y n−1 + hλy n (4.99) Solving for y n , 1 y n = 1 − hλ y n−1 (4.100) The absolute stability requirement |y n /y n−1 | < 1 gives or (substituting z = hλ = x + iy, |1 − hλ| ≥ 1 (4.101) (1 − x) 2 + y 2 ≥ 1 (4.102) Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 4. IMPROVING ON EULER’S METHOD 81 Thus the region of absolute stability for the Backwards Euler Method encompasses the entire complex plane outside the circle of radius 1 centered at the point (1, 0), as illustrated in figure 4.6. Figure 4.6: The gray area indicates the region of absolute stability for the Backwards Euler Method. The Backwards Euler Method is an example of an implicit method, because it contains y n implicitly. In general it is not possible to solve for y n explicitly as a function of y n−1 in equation 4.98, even though it is sometimes possible to do so for specific differential equations. Thus at each mesh point one needs to make some first guess to the value of y n and then perform some additional refinement to improve the calculation of y n before moving on to the next mesh point. A common method is to use fixed point iteration on the equation where k = y n−1 . The technique is summarized here: y = k + hf(t, y) (4.103) • Make a first guess at y n and use that in right hand side of 4.98. A common first guess that works reasonably well is y (0) n = y n−1 (4.104) • Use the better estimate of y n produced by 4.98 and then evaluate 4.98 again to get a third guess, e.g., y (ν+1) n = y n−1 + hf(t n , y (ν) n ) (4.105) • Repeat the process until the difference between two successive guesses is smaller than the desired tolerance. c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
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The Computable Differential Equatio
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Contents 1 Classifying The Problem
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CONTENTS v Timeline on Computable D
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Chapter 1 Classifying The Problem 1
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CHAPTER 1. CLASSIFYING THE PROBLEM
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Chapter 2 Successive Approximations
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CHAPTER 2. SUCCESSIVE APPROXIMATION
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CHAPTER 6. LINEAR MULTISTEP METHODS
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CHAPTER 6. LINEAR MULTISTEP METHODS
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Chapter 7 Delay Differential Equati
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CHAPTER 7. DELAY DIFFERENTIAL EQUAT
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Chapter 8 Boundary Value Problems 8
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CHAPTER 8. BOUNDARY VALUE PROBLEMS
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Chapter 9 Differential Algebraic Eq
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CHAPTER 9. DIFFERENTIAL ALGEBRAIC E
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Bibliography [1] Ascher, Uri M., Ma
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