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

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184 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS There is no solution for (x n , y n ) in terms of the variables at earlier times. This can be seen be rewriting the system as x n − t n y n = h(x n−1 − t n y n−1 ) (9.171) x n − t n y n = f(t n ) (9.172) or in matrix form ( ) ( ) 1 −tn xn = 1 −t n y n ( ) h(xn−1 − t n y n−1 ) f(t n ) The matrix of coefficients is singular hence there is no solution. (9.173) 9.6 BDF Methods for DAEs While the numerical solution of initial value problems for ordinary differential equations has a long history going back over one hundred years, the numerical solution of DAEs is mostly still an open research issue (see [5] for the more common methods) with significant interest only dating back to the 1970’s. Since many of the scientific problems of interest turn out to be stiff, early research focused first on BDF methods and later on some RK methods that are known to be better for stiff problems. The first DAE algorithms published were BDF methods. 5 The solution of linear and index-1 systems are now understood but methods for higher index systems still form the basis of much current methods. In general variable step sizes can cause problems with DAEs. For example, it is not possible to solve an index-3 system using implicit Euler with variable step-size because of the following theorem because it will have O(1) errors. Theorem 9.11. The global error for a k-step BDF applied to an index-ν system is O(h n ) where n = min(k, k − ν + 2) Furthermore, all multistep (as well as Runge-Kutta) methods are unstable for higher index (≥ 3) systems, and in some cases fail for specific index-1 and index-2 systems. Some convergence results for semi-implicit and fully-implicit index-1 systems, semi-explicit index-2 systems, and index-3 systems in Hessenberg form have been worked out. We can write the general k-step BDF method for a scalar ODE (see 6.105) as k∑ a j y k−j = hb 0 Dy n (9.174) j=0 where D is the differential operator Dy = y ′ . Then the general BDF method becomes 0 = F (t n , y n , Dy n ) (9.175) 5 C.W.Gear (1971) “Simultaneous Numerical Solution of Differential-Algebraic Equations,” IEEE Transactions on Circuit Theory, 18:89-95 Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 185 where Dy n = 1 hb 0 k∑ a j y k−j (9.176) j=0 Theorem 9.12. Let Ay ′ + By = f(t) (9.177) be a linear constant coefficient DAE of index-ν with regular pencil λA + B, and let 9.174 be a k-step (where k < 7) constant step-size BDF method. Then the BDF method converges for this system to order O(h k ) after (ν − 1)k + 1 steps. Proof. The BDF method for the linear equation is where Dy n is given by 9.176. matrices P and Q, where P AQ = ADy n + By n = f n (t) (9.178) Since the pencil is regular there are nonsingular ( ) I 0 ; P BQ = 0 N with N a nilpotent matrix of nilpotentcy ν, so that ( ) C 0 0 I (9.179) P AQDy n + P BQy n = P f n (t) (9.180) ( ) ( ) I 0 C 0 Dy 0 N n + y 0 I n = P f n (t) (9.181) Partitioning y n = ( un Multiplying out the matrix expressions, v n ) ; P f n (t) = ( ) gn (t) h n (t) (9.182) Du n + Cu n = g n (t) (9.183) NDv n + v n = h n (t) (9.184) Equation 9.183 is a pure ODE expression for the BDF method that we already know is convergent of order-k. Equation 9.184 can be rewritten as (I + ND)v n = h n (t) (9.185) From equation 9.59 and therefore ∑ν−1 (I + ND) −1 = (−1) j N j D j (9.186) j=0 ∑ν−1 v n = (−1) j N j D j h n (t) (9.187) j=0 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 6 Linear Multistep Methods
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CHAPTER 6. LINEAR MULTISTEP METHODS
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Page 205 and 206: Bibliography [1] Ascher, Uri M., Ma
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