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math572_hw4

# math572_hw4 - MATH 572 Assignment#4 Numerical Methods for...

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MATH 572 Numerical Methods for Scientific Computing II Winter 2005 Assignment #4 due : Thursday , March 10 1. Consider the 2-step BDF scheme, u n + 1 2 2 u n = hf ( u n ). a) Find the characteristic roots ζ 1 ( h ) , ζ 2 ( h ) for the test equation and plot them using Matlab over the interval - 10 0. b) Show analytically that the negative real axis is contained in the region of absolute stability. (Note: the scheme is actually A-stable, but it is not required to show that here.) 2. The Lorenz system is defined by y 1 y 2 y 3 = σ ( y 2 - y 1 ) ry 1 - y 2 - y 1 y 3 y 1 y 2 - by 3 . These equations were originally derived as a model for thermal convection; the variables represent the temperature, density, and velocity in a certain fluid flow. It was discovered by numerical computations that the parameters σ = 10 , b = 8 / 3 , r = 28 yield a system with chaotic dynamics. Solve this system with initial conditions y 1 (0) = 0 , y 2 (0) = 1 , y 3 (0) = 0 up to time t = 100 using Matlab’s ode45 solver. (Part of the exercise is to read the online
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