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# set17 - \$ Set 17 Ordinary Differential Equations Linear...

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a39 a38 a36 a37 Set 17: Ordinary Differential Equations: Linear Multistep Methods Kyle A. Gallivan Department of Mathematics Florida State University Foundations of Computational Math 2 Spring 2011 1

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a39 a38 a36 a37 Sources U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations, SIAM, 1998. J. D. Lambert, Numerical Methods for Ordinary Differential Systems, Wiley 1991, 1973. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, 1973. R. Skeel, Numerical Differential Equations Class Notes, University of Illinois, 1979. 2
a39 a38 a36 a37 Linear One-step Methods general form α 0 y n + α 1 y n 1 = h ( β 0 f n + β 1 f n 1 ) forward Euler y n = y n 1 + hf n 1 y n y n 1 = hf n 1 α 0 = 1 , α 1 = 1 , β 0 = 0 , β 1 = 1 backward Euler y n = y n 1 + hf n y n y n 1 = hf n α 0 = 1 , α 1 = 1 , β 0 = 1 , β 1 = 0 Trapezoidal rule y n = y n 1 + h 2 ( f n + f n 1 ) y n y n 1 = h ( 1 2 f n + 1 2 f n 1 ) α 0 = 1 , α 1 = 1 , β 0 = 1 / 2 , β 1 = 1 / 2 3

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a39 a38 a36 a37 General Form of Linear Multistep Methods Assume h n = h and let f n = f ( t n , y n ) where y n is a point on the numerical solution. k step Linear multistep methods are of the form: k summationdisplay j =0 α j y n j = h k summationdisplay j =0 β j f n j N h [ y n ] = k j =0 α j y n j h k summationdisplay j =0 β j f n j α 0 negationslash = 0 y n k and/or f n k involved ⇒ | α k | + | β k | negationslash = 0 Initial conditions must be specified y 0 , . . . , y k 1 4
a39 a38 a36 a37 Questions How does the form of linear multistep methods affect the following? Derivation of methods Consistency of methods 0 and absolute stability of methods Convergence of methods 5

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a39 a38 a36 a37 Derivations Various derivations of these methods are possible depending on the family. algebraic constraints difference operator calculus interpolation and integration interpolation and differentiation 6
a39 a38 a36 a37 Adams Methods y ( t n ) = y ( t n 1 ) + Z t n t n - 1 f ( t, y ( t )) dt Adams-Bashforth – explicit methods, k - step, order k let P ( t ) interpolate f n 1 , . . . , f n k Define the integration constant so that P ( t n 1 ) = y n 1 The method is given by y n = P ( t n ) Adams-Moulton – implicit methods, k - step, order k + 1 let P ( t ) interpolate f n , f n 1 , . . . , f n k Define the integration constant so that P ( t n 1 ) = y n 1 The method is given by y n = P ( t n ) 7

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a39 a38 a36 a37 Example Forward Euler: P ( t ) = f n 1 P ( t ) = tf n 1 + c y n 1 = P ( t n 1 ) y n 1 = t n 1 f n 1 + c c = y n 1 t n 1 f n 1 y n = P ( t n ) = t n f n 1 + y n 1 t n 1 f n 1 = y n 1 + hf n 1 8
a39 a38 a36 a37 Example Trapezoidal rule: P ( t ) = ( t t n 1 ) ( t n t n 1 ) f n ( t t n ) ( t n t n 1 ) f n 1 P ( t ) = 1 2 h bracketleftbig ( t t n 1 ) 2 f n ( t t n ) 2 f n 1 bracketrightbig + c y n 1 = P ( t n 1 ) c = y n 1 + h 2 f n 1 y n = 1 2 h bracketleftbig ( t n t n 1 ) 2 f n bracketrightbig + y n 1 + h 2 f n 1 = y

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• Spring '11
• gallivan
• yn, Numerical differential equations, Numerical ordinary differential equations, Linear multistep method, Linear Multistep Methods

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