B2 Consider the linear ?rst order ODE y?=-Ay, y(0)=yo>0 for ?xed A > 0. Let (yn)n be at sequence de?ned by a difference equation, approximating the value of the solution to this ODE at equidistant points in time, i.e., yn % y(nh) for h > 0 given. We start the sequence with the initial condition yo from the ODE problem. We de?ne two notions of stability for the difference equation de?ning yn: A?stable lim,,_>00 yn = Yc>0 where Yc>0 is the value of the solution y for t ?> oo (in the above case Y00 = 0). So yn matches the long time behavior of the solution to the ODE. M?stable yn is positive and the sequence is monotone decreasing (which is also true for the exact solution of the ODE). So if the DE satis?es these stability properties the sequence (yn)n matches the behavior of the exact solution y. In many cases these stability properties hold only if h is small enough (this is called conditionally stable). In this case one can ?nd a ho depending on A so that stability holds for h < ho. If some approximation method satis?es some stability property for all h one says that the method is unconditionally stable. Hint: It might help in the following to de?ne p = hA and to ?nd p0 so that stability holds for p < p0 this implies then ho = ?3?0. As you can see from this, conditionally stable approximations can require very small step sizes h if A is large (if A is large one says that the ODE is sti??). 1. Find he so that the explicit Euler method applied to the above linear ODE is A?stable for h < ho. Find he so that the method is M?stable for h < ho. 2. Show that the implicit Euler method applied to the above linear ODE is both unconditionally A?stable and unconditionally M?stable. 3. Now conider the following second order difference equation for computing approximations yn: yn+2 + ayn+1 + byn : Chf(yn+2) '
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