chw4 - solution Is it possible to get higher accuracy by...

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Math 128a, fall 2011, Chorin, computer homework 4 (This computer homework focuses on things that do NOT work. Why they do not work has not yet been discussed. I want you to see that things do not work unless one does the right thing). 1. Write a computer program that solves the equation y 0 = f ( x,y ) ,y (0) = a , ( a is a given number), by the centered scheme u 0 = a,u 1 = u 0 + hf (0 ,u 0 ) , and for i > 1, u i +1 = u i - 1 + 2 hf ( x i ,u i ), where h is the mesh size and x i = ih . Use it to solve the equations y 0 = - 2 y,y 0 = 2 y , in both cases with the condition y (0) = 1. Follow the solution up to x = 2, with h = 0 . 1 and h = 0 . 5. In each of the 4 cases, plot the error (numerical solution minus exact solution) and the relative error (error divided by the exact
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Unformatted text preview: solution). Is it possible to get higher accuracy by extrapolation? 2. Check that y = ( y i +1 + (3 / 2) y i-3 y i-1 + (1 / 2) y i-2 ) / (3 h ) + O ( h 3 ) , where y i = y ( x i ). Write a program that implements the resulting scheme for solving y = f ( x,y ) ,y (0) = 1: u = 1 ,u 1 = u + hf (0 , 1) ,u 2 = u + 2 hf ( h,u 1 ) , and for i > 2, u i +1 =-(3 / 2) u i +3 u i-1-(1 / 2) u i-2 +3 f ( x i ,u i ). Use it to solve y =-y,y (0) = 1, for 0 < x ≤ 2, with h = . 1 ,h = . 2. Comment on what you observe. 1...
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