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Unformatted text preview: Test I Solutions, MAT 421 (1) Write down the lowest order interpolating polynomial P ( x ) that goes through the three points (- 1 , 6), (0 , 1), (2 , 3) using Lagrange interpolation. P ( x ) = N k =0 j = k ( x- x j ) ( x k- x j ) f k P ( x ) = 2 x 2- 3 x + 1 (2) For du/dt = f ( u ), prove that the backward Euler method u n +1 = u n + Δ tf ( u n +1 ) is first-order accurate, using the definition of the local trun- cation error LTE = Δ tτ . Include the constants in the global error τ . u ( t + Δ t ) = u + Δ tu ( t + Δ t ) + Δ tτ Taylor expanding, we get u + Δ tu + Δ t 2 u / 2 + · · · = u + Δ tu + Δ t 2 u + · · · + Δ tτ or τ =- Δ tu / 2 + O (Δ t 2 ). (3) (a) For du/dt = au , a < 0, prove that the TR method u n +1 = u n + Δ t 2 ( f ( u n ) + f ( u n +1 )) is A-stable. (b) Is TR L-stable? Hint: Find lim Δ t →∞ G , where G is the growth factor. The growth factor for TR for du/dt = au is u n +1 = 1 + a Δ t/ 2 1- a Δ t/ 2 u n ≡ Gu n For the model problem (...
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This note was uploaded on 03/11/2012 for the course MAT 421 taught by Professor Staff during the Fall '11 term at ASU.
- Fall '11