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

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190 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS where [w(i)] i=1,...,s represents the vector of components indexed by i. Define ⎛ ⎞ g(t n−1 + c 1 h) − g(t n−1 ) G = 1 g(t n−1 + c 2 h) − g(t n−1 ) ⎜ ⎟ h ⎝ . ⎠ g(t n−1 + c s h − g(t n−1 ) (9.221) so that Then from equation 9.212, V ′ = A −1 G (9.222) v n = v n−1 + hb T V ′ (9.223) = g(t n−1 ) + hb T A −1 G (9.224) The local error is thus d n = g(t n−1 ) + hb T A −1 G − g(t n ) (9.225) ) = − (hg ′ + h2 2 g′′ + h3 6 g′′′ + · · · (9.226) ⎛ c 1 hg ′ + c2 1 h2 2 g′′ + c3 1 h3 ′′′ g + · · · + b T A −1 ⎜ ⎝ . ⎟ (9.227) c s hg ′ + c2 s h2 2 g′′ + c3 2 h3 ′′′ ⎠ g + · · ·⎞ Definition 9.14. The algebraic order of an IRK method is equal to k if d n = O(h k+1 ) for all components. Theorem 9.16. The algebraic order of an IRK method equal to k if and only if where b T A −1 c j = 1, for j = 1, . . . , k (9.228) We state the remaining results without proof. c j = (c j 1 , . . . , cj s) T (9.229) Definition 9.15. A perturbation (z n , δ 0 , . . . , δ s+1 ) of the method for y n , ⎛ ⎞ s∑ F ⎝t n−1 + c i h, y n−1 + h a ij Y j, ′ Y i ′ ⎠ = 0 (9.230) j=1 s∑ y n = y n−1 + h b i Y i ′ (9.231) i=1 Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 191 is given by ⎛ ⎞ s∑ F ⎝t n−1 + c i h, z n−1 + h a ij Z ′ j + δ n (i) , Z ′ ⎠ i = 0 (9.232) j=1 z n = z n−1 + h s∑ i=1 b i Z ′ i + δ s+1 n (9.233) where z 0 = y 0 + δ s+1 0 . Definition 9.16. Let (z n , δ 0 , . . . , δ s+1 ) be a perturbation of the IRK method y n for a particular differential algebraic equation F (t, y, y ′ ) = 0, with |δ n (i) | ≤ ∆, i = 1, . . . , s + 1, for some number ∆. Then if |z n − y n | ≤ K 0 ∆ for 0 < h ≤ h 0 , for some constants K 0 and h 0 that depend only on the method and the differential algebraic equation, then we say the method is strictly stable for the DAE. Theorem 9.17. An IRK method with A nonsingular is strictly stable for a linear constant-coefficient index-1 differential algebraic equation iff and only iff where 1 is as defined in equation 5.211. |1 − b T A −1 1| < 1 (9.234) Recall from that the stability function R(z) = y n /y n−1 satisfies Taking the limit as |z| → ∞, R(z) = 1 + zb T (I − zA) −1 1 (9.235) lim R(z) = 1 − |z|→∞ bT A −1 1 (9.236) From theorem 9.17 implies that an implicit Runge-Kutta method is stable when r = lim |R(z)| < 1 (9.237) |z|→∞ Definition 9.17. k c is called the constant coefficient order of an implicit Runge- Kutta method if the method converges with global error O(h kc ) for all linear constant coefficient index-one systems. Theorem 9.18. Let A, b, c be an implicit Runge-Kutta method that is strictly stable, with constant-coefficent order k c , algebraic order k a , and differential order k d . Then k c = min(k a + 1, k d ) (9.238) 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 3 Approximate Solutions 3.1
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CHAPTER 3. APPROXIMATE SOLUTIONS 45
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Chapter 5 Runge-Kutta Methods 5.1 T
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CHAPTER 5. RUNGE-KUTTA METHODS 97 F
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CHAPTER 5. RUNGE-KUTTA METHODS 101
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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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Page 205 and 206: Bibliography [1] Ascher, Uri M., Ma
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