math572_hw3 - MATH 572 Numerical Methods for Scientific...

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Unformatted text preview: MATH 572 Numerical Methods for Scientific Computing II Winter 2005 Assignment #3 due : Tuesday, February 22 1. Solve the difference equation for u n . Check and make sure your answer satisfies the initial conditions and difference equation. a) u n- 2 u n- 1- 3 u n- 2 = 0 , u = 0 , u 1 = 1 b) u n- 3 u n- 1 + 3 u n- 2- u n- 3 = 0 , u = 1 , u 1 = 0 , u 2 =- 3 2. Consider the following 2-step methods for y = f ( y ), scheme 1 : u n- u n- 2- h 3 ( f ( u n ) + 4 f ( u n- 1 ) + f ( u n- 2 )) = 0 (Milne’s method ) scheme 2 : u n + 4 u n- 1- 5 u n- 2- h ( 4 f ( u n- 1 ) + 2 f ( u n- 2 ) ) = 0 Answer the following questions for each scheme. a) Find ρ ( ζ ) , σ ( ζ ). Is the scheme consistent? Is the root condition satisfied? b) Find the local truncation error in the form τ n = Cy ( r +1) ( t ) h r +1 + O ( h r +2 ). c) Consider the equation y = λy . Find the characteristic roots and show that they satisfy scheme 1 : ζ 1 ( h ) = e hλ + O ( h 5 ) , ζ 2 ( h ) =- e- hλ/ 3 + O ( h 3 ) scheme 2 :...
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