# hw6 - K ’s and check that u i 1 can be written as the...

This preview shows page 1. Sign up to view the full content.

Math 128a, fall 2011, Chorin, theory homework 5, due the week of Oct. 17. 1. Solve (analytically) the following initial initial value problems for i > 1: (i) u i +1 - u i + 0 . 25 u i - 1 = 0 ,u 0 = 1 ,u 1 = 1. (ii) u i +1 + u i - 1 = 0 ,u 0 = 1 ,u 1 = - 1 . 2. Which of the following diﬀerence schemes have only solutions that are bounded for all i,h such that ih = x , where x is a ﬁxed positive number: (i) u i +1 + u i - 1 = hu i , (ii) u i +1 - 2 u i + u i - 1 = 0 . 3. Which of the following sequences remain bounded as i → ∞ while ih = x , with x constant: (i) u i = ( . 9) i , (ii) u i = (1 + h ) i , (iii) u i = (1 + h ) i . 4. Consider the standard fourth-order RK method. Apply it to solving the the equation y 0 = ay,y (0) = 1. calculate (analytically) the various
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: K ’s, and check that u i +1 can be written as the beginning of a Taylor series for y i +1 around x i . Deduce the order of accuracy of the scheme in this case from this calculation. 5. Consider a diﬀerence scheme of the form a 1 u i +1 + a u i + a-1 u i-1 + a-2 u i-2 = hf ( x i ,u i ), which is a consistent approximation of y = f ( x,y ) ( a 1 ,a ,a i-1 , etc. are constants). Consider the polynomial P ( r ) = a 1 r i +1 + a r i + a-1 r i-1 + a-2 r i-2 . This polynomial always has r = 1 as a root. Explain why. 1...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online