math572_hw2 - MATH 572 Numerical Methods for Scientific...

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Unformatted text preview: MATH 572 Numerical Methods for Scientific Computing II Winter 2005 hw#2 due : Tuesday , February 8 1. Find the local truncation error for the backward Euler method u n +1 = u n + hf ( u n +1 ). 2. Consider the midpoint method u n +1 = u n + hf ( u n + h 2 f ( u n )) applied to the test equation y = y . a) Assume that is a negative real number. Show that h is contained in the region of absolute stability if and only if- 2 h 0. b) Show that no imaginary value of h is contained in the region of absolute stability. 3. Consider the trapezoid method u n +1 = u n + h 2 ( f ( u n ) + f ( u n +1 )). Find the local truncation error and show that the scheme is A-stable. 4. Consider the initial value problem y 1 y 2 =- 11 9 9- 11 y 1 y 2 , y 1 y 2 = 1 . 1 . 2 . a) Find an analytic expression for the exact solution of the problem and also for the numerical solutions given by the forward Euler and backward Euler methods. In each case show analytically that lim h u 1 ,n = y 1 ,n...
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