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# solhw10 - Homework 10 Foundations of Computational Math 2...

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Unformatted text preview: Homework 10 Foundations of Computational Math 2 Spring 2011 Problem 10.1 Consider the following linear multistep method: y n =- 4 y n − 1 + 5 y n − 2 + h (4 f n − 1 + 2 f n − 2 ) The method is not 0-stable. 10.1.a . Determine, p , the order of consistency of the method. 10.1.b . Determine the coefficient, C p +1 , in the discretization error d n . 10.1.c . Consider the application of the method to y ′ = 0 with y = 0 and y 1 = ǫ , i.e., a perturbed initial condition. Show that | y n | → ∞ as n → ∞ , i.e., the numerical method is unstable. Solution: We are given the linear multistep method: y n =- 4 y n − 1 + 5 y n − 2 + h (4 f n − 1 + 2 f n − 2 ) The method is not 0-stable since ρ ( ξ ) = ξ 2 + 4 ξ- 5 = ( ξ- 1)( ξ + 5) which violates the root condition for 0-stability of a linear multistep method. The order of consistency can be checked either by using the formulae for C i ’s given in the notes or by inserting Taylor expansions and simplifying. We have α = 1 , α 1 = 4 , α 2 =- 5 β = 0 , β 1 = 4 , β 2 = 2 C = C 1 = C 2 = C 3 = 0 C 4 = 4 1 24- 5 16 24 + 4 1 6 + 2 8 2 = 1 6 d n = 1 6 h 3 y (4) ( t n ) + O ( h 4 ) 1 Applying the method to y ′ = 0 with y = 0 and y 1 = ǫ we get the following: y n =- 4 y n − 1 + 5 y n − 2 = γ 1 (1) n + γ 2 (- 5) n γ 1 = ǫ 6 γ 2 =- ǫ 6 y n = ǫ 6- ǫ 6 (- 5) n = ǫ 6 (1 + (- 1) n +1 (- 5) n ) = μ ( n ) ǫ Clearly, | μ ( n ) | → ∞ . The growth is easily seen in the first few terms: y = 0 y 1 = ǫ y 2 =- 4 ǫ y 3 = 21 ǫ Problem 10.2 Consider the following linear multistep method: y n = y n − 2 + h 3 ( f n + 4 f n − 1 + f n − 2 ) The method is 0-stable but it is weakly stable. 10.2.a . Determine the discretization error d n . 10.2.b . Consider the application fo the method to y ′ = λy . Write the recurrence that yields y n . 10.2.c . Show that | y n | → ∞ as n → ∞ , i.e., the numerical method is unstable. Solution: For y n = y n − 2 + h 3 ( f n + 4 f n − 1 + f n − 2 ) we have C = C 1 = C 2 = C 3 = C 4 = 0 C 5 =- 1 90 2 So the method is order p = 4....
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## This note was uploaded on 11/10/2011 for the course MAD 5404 taught by Professor Gallivan during the Spring '11 term at FSU.

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solhw10 - Homework 10 Foundations of Computational Math 2...

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