MATH 572Numerical Methods for Scientific Computing IIWinter 2005Assignment #4due : Thursday , March 101. Consider the 2-step BDF scheme,∇un+12∇2un=hf(un).a) Find the characteristic rootsζ1(h), ζ2(h) for the test equation and plot them usingMatlab over the interval-10≤hλ≤0.b) Show analytically that the negative real axis is contained in the region of absolutestability. (Note: the scheme is actually A-stable, but it is not required to show that here.)2. The Lorenz systemis defined byy1y2y3=σ(y2-y1)ry1-y2-y1y3y1y2-by3.These equations were originally derived as a model for thermal convection; the variablesrepresent the temperature, density, and velocity in a certain fluid flow. It was discovered bynumerical computations that the parametersσ= 10, b= 8/3, r= 28 yield a system withchaotic dynamics. Solve this system with initial conditionsy1(0) = 0, y2(0) = 1, y3(0) = 0up to timet= 100 using Matlab’sode45solver. (Part of the exercise is to read the online
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