hmwk_7_2010_solutions

hmwk_7_2010_solutions - KEY Provided courtesy of Jake T...

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Unformatted text preview: KEY Provided courtesy of Jake T. Lussier December 8, 2010 Part 1 Time U D Δ x Δ t C P Stability Accuracy Peak Phase Wiggles Peak Stepper m/s m 2 /s m s Damping Explicit 1.0 0.0 1.0 0.1 0.1 ∞ Unstable Very Inacc. Lagged Major None Explicit 1.0 0.0 1.0 1.0 1 ∞ Rap. Un. Very Inacc. Lagged Major None Explicit 1.0 0.0 0.4 0.04 0.1 ∞ Slow. Un. Acc. Lagged Minor None Explicit 1.0 0.0 0.4 0.4 1 ∞ Rap. Un. Very Inacc. Lagged Major None Implicit 1.0 0.0 1.0 0.1 0.1 ∞ Stable Inacc. Slight Lagged Minor Damped Implicit 1.0 0.0 1.0 1.0 1 ∞ Stable Inacc. Lagged None Highly D. Implicit 1.0 0.0 0.4 0.04 0.1 ∞ Stable Inacc. Acc. None Damped Implicit 1.0 0.0 0.4 0.4 1 ∞ Stable Inacc. Acc. None Highly D. Crank- 1.0 0.0 1.0 0.1 0.1 ∞ Stable Inacc. Lagged Minor Damped Nicolson Crank- 1.0 0.0 1.0 1.0 1 ∞ Stable Inacc. Lagged Minor Damped Nicolson Crank- 1.0 0.0 0.4 0.04 0.1 ∞ Stable Very Acc. Acc. None None Nicolson Crank- 1.0 0.0 0.4 0.4 1 ∞ Stable Very Acc. Acc. V. Slight V. Slight Nicolson Table 1: Table for Part 1 (a) We used ADDIF.m to solve, plot, and compare each of the listed cases to the analytical solution. The results are shown in Figure 1. Qualitative comments on the stability, accuracy, peak phase, wiggles, and peak damping are also shown in Table 1. (b) The Courant numbers for all cases are shown in Table 1. Results may be either stable or unstable, accurate or inaccurate for either C = 0.1 or C = 1.0. However, in general, a Courant value of 0.1 seems to yield more accurate and stable solutions than a Courant value of 1.0. (c) In looking at the plots in Figure 1, and in looking at the associated error norms, we can see that Crank-Nicolson is the most accurate, followed by implicit, followed by explicit. (d) The grid resolution must be small relative to the standard deviation of the plume so that the approximation can capture the plume. Higher grid resolutions clearly give better results for C-N and implicit. 1 (a) L 2 = 0.178433, L inf = 0.986083 (b) L 2 = 3.64014e+025, L inf = 1.989e+026 (c) L 2 = 0.0340095, L inf = 0.185558 (d) L 2 = 3.85631e+076, L inf = 2.36692e+077 (e) L 2 = 0.0454272, L inf = 0.348007 (f) L 2 = 0.0927528, L inf = 0.69894 (g) L 2 = 0.0198214, L inf = 0.165994 (h) L 2 = 0.072011, L inf = 0.552901= 0....
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This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.

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hmwk_7_2010_solutions - KEY Provided courtesy of Jake T...

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