solhw10

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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 = , 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 ( ) = 2 + 4 - 5 = ( - 1)( + 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 = 1 , 1 = 4 , 2 =- 5 = 0 , 1 = 4 , 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 = we get the following: y n =- 4 y n 1 + 5 y n 2 = 1 (1) n + 2 (- 5) n 1 = 6 2 =- 6 y n = 6- 6 (- 5) n = 6 (1 + (- 1) n +1 (- 5) n ) = ( n ) Clearly, | ( n ) | . The growth is easily seen in the first few terms: y = 0 y 1 = y 2 =- 4 y 3 = 21 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 = y . 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....
View Full Document

Page1 / 6

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online