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

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114 CHAPTER 5. RUNGE-KUTTA METHODS Example 5.5. Find the stability function for the classical fourth-order Runge-Kutta method 0 0 1/2 1/2 0 1/2 0 1/2 0 (5.226) 1 0 0 1 0 1/6 1/3 1/3 1/6 Solution. ⎛ ⎞ 1 0 0 0 I − zA = ⎜−z/2 1 0 0 ⎟ ⎝ 0 −z/2 1 0⎠ (5.227) 0 0 −z 1 Therefore det(I − zA) = 1 and ⎛ ⎞ ⎛ ⎞ 1 0 0 0 1 2 2 1 I − zA + z1b T = ⎜−z/2 1 0 0 ⎟ ⎝ 0 −z/2 1 0⎠ + z ⎜1 2 2 1 ⎟ 6 ⎝1 2 2 1⎠ (5.228) 0 0 −z 1 1 2 2 1 ⎛ ⎞ 1 + z/g z/3 z/3 z/6 = ⎜ −z/3 1 + z/3 z/3 z/6 ⎟ ⎝ z/6 −z/6 1 + z/3 z/6 ⎠ (5.229) z/6 z/3 −2z/3 1 + z/6 Therefore det(I − zA + z1b T ) = 1 + z + z2 2 + z3 6 + z4 24 R(z) = 1 + z + z2 2 + z3 6 + z4 24 (5.230) (5.231) The formulas given by these theorems are examples of Padé Approximants. More specifically, a Padé Approximant is P m/n is a rational function P m/n (t) = P m(t) P n (t) (5.232) where P m and P n are polynomials of degrees m and n, respectively. The Padé Approximant is the best approximation of function by a rational function of a given order; it is essentially the Rational-function generalization of a Taylor approximation. We observe that not only is the result a rational function, it is a polynomial. This result is true in general for all explicit Runge-Kutta methods. Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 5. RUNGE-KUTTA METHODS 115 Theorem 5.4. For every Runge-Kutta method there exists a rational function P ν/ν such that y n = [ P ν/ν (z) ] n (5.233) Proof. Start with R(z) = 1 + zb T (I − zA) −1 1 (5.234) which is a result we have already demonstrated. Then (I − zA) −1 = adj(I − zA) det(I − zA) (5.235) where adj(M) is the adjunct matrix of M, [adj(M)] ij = (−1) i+j minor(M ji ) (5.236) We observe that each entry in the adjunct matrix is linear in z, hence each of the principal minor determinants is a polynomial in z of order ν − 1. Therefore zb T adj(I − zA) −1 1 (5.237) is a polynomial of degree ν. Hence zb T (I − zA) −1 1 is the quotient of two polynomials, each of order ν, which immediately yields the desired result. Corollary 5.1. For an explicit Runge-Kutta method, y n = [P ν ] n , a polynomial of order ν. Proof. If the method is explicit then A is strictly lower triangular; hence det(I − zA) = 1. Corollary 5.2. Explicit Runge-Kutta methods may not be A-stable. Proof. A non-constant polynomial cannot be uniformly bounded. Theorem 5.5. Let R(z) be any rational functions. Then |R(z)| < 1 for all z ∈ C − if and only if both of the following conditions hold: 1. All the poles of R(z) have positive real parts; and 2. |R(it)| ≤ 1 for all t ∈ R. Theorem 5.6. R(z) = e z + O(z ν 1 ) for any Runge-Kutta method for order ν. Proof. By definition, y n = R(z)y n−1 ; for the test equation, y(t n ) = y n = e z y n−1 . Theorem 5.7. For any two integers m, n there is a Padé Approximant to the exponential for the stability function such that P m/n (t) = P m,n(t) Q m,n (t) (5.238) 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 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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