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Math 480 Spring 2010 Exercises II Due date, April 7th, 2010 Consider the linear multi-step method U n+2 U n = h(fn + fn+1 + fn+2 ) () for solving the...

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Math 480 Spring 2010 Exercises II Due date, April 7th, 2010 Consider the linear multi-step method U n +2 - U n = h ( αf n + βf n +1 + γf n +2 ) ( ? ) for solving the ordinary differential equation u 0 ( t ) = f ( u ( t )) with u ( t 0 ) = u 0 . 1. For what values of α,β , and γ is this method consistent? 2. For what values of α,β , and γ is ( ? ) explicit with O ( h 2 ) truncation error? 3. For what values of α,β , and γ does ( ? ) have O ( h 3 ) truncation error? (The resulting method(s) need not be explicit.) 4. For what values of α,β , and γ is ( ? ) zero stable? What does this imply about the convergence of solution estimates as h 0? 5. Suppose we take α = 2 and β = γ = 0, and apply this method to the equation u 0 ( t ) = - u ( t ). For which values of h will the approximate solution { U n } exponentially decay as n → ∞ ? Solving a difference equation 1. Determine the general solution to the linear difference equation 2 U n +3 - 5 U n +2 + 4 U n +1 - U n = 0. Hint: One root of the characteristic polynomial is at ζ = 1. 2. Determine the solution to this difference equation with the start- ing values U 0 = 11, U 1 = 5, and U 2 = 1. What is U 10 ? 3. Consider the LMM 2 U n +3 - 5 U n +2 + 4 U n +1 - U n = k ( β 0 f ( U n ) + β 1 f ( U n +1 )) . For what values of β 0 and β 1 is local truncation error O ( k 2 )? 4. Suppose you use the values of β 0 and β 1 just determined in this LMM. Is this a convergent method? 1
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