hw7 - u i +1 = u i + hf i , (ii) by the implicit Euler...

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Math 128a, fall 2011, Chorin, theory homework 7, due the week of Oct. 31. 1. Which of the following schemes are stable:(i) u i +1 = u i + hf i ; (ii) u i +1 = u i + hf i - 1 ; (iii) u i +1 = 2 u i + hf i ; (iv) u i +1 = u i +7; (v) u i +1 = (1+ h ) u i . In all cases, f i means f ( x i ,u i ). 2. Suppose are solving the equation y 0 = f ( x,y ) by the scheme u i +1 = u i - 3 + h ( a 1 f i - 1 + a 3 f i - 3 + a 4 f i - 4 ). Find a 1 ,a 3 ,a 4 so that the local truncation error is O ( h 3 ). 3. Find the order of the local truncation error for the following predic- tor/corrector pairs: (i) u * i +2 = u i +2 hf i +1 ; u i +2 = u i + h ( f i / 3+4 f i +1 / 3+ f ( x i +2 ,u * i +2 ) / 3); (ii) u i +1 = u i + hf i , u i +1 = u i +( h/ 2)( f i + f ( x i +1 ,u * i +1 )) . 4. Consider the equation y 0 = - 1000 y,y (0) = 1. The solution is y = e - 1000 x and rapidly tends to zero. Suppose you want to solve this equation (i) by the Euler scheme
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Unformatted text preview: u i +1 = u i + hf i , (ii) by the implicit Euler scheme u i +1 = u i + hf i +1 , each with h = 0 . 01. In each case you can nd the exact solution of the dierence equation in the form u i = i for some appropriate ; plot by hand the solutions in each case for i = 1 , 2 , 3 , 4. How do you reconcile what you see what the fact that both schemes are convergent? which of the two is better (or more precisely, less horrible) in this case? what is the absolute value of the error at x = . 1 in each case? 1...
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