Lecture 5

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Unformatted text preview: rther suppose 40 that the starting values are O(h) (so that the starting values converge to y0 = 0). The two solutions of the di erence equation corresponding to 2 and 3 are c2h n n c3hn 3 : 2 (The coe cients c2 and c3 have been scaled to re ect the O(h) size of the solution.) The analysis is simpli ed by letting Y (t) be a continuous function that interpolates to yj , j = 0 1 : : : . Thus, Y2(t) = c2 h t=h Y3(t) = c3t 2 t=h t = nh: 3 For xed t, the solutions Y2(t) and Y3(t) are unbounded as h ! 0 thus, the solution cannot be convergent and we have a contradiction. Therefore, the LMM must be stable. Next consider the IVP y0 = 0, y(0) = 1, which has the solution y(t) = 1. With the present notation, the di erence equation (5.1.2) becomes k X i=0 Convergence implies that i Y (t ; ih) = 0: lim Y (t ; ih) = 1 h!0 or, using (5.6.7b), that (1) = 0. Finally, consider the IVP y0 = 1, y(0) = 0, which has the solution y(t) = t. The di erence equation (5.1.2) for this problem is k X i=0 i Y (t ; ih) = h k X i=0 i: Convergence would imply that Y (t ; ih) = t ; ih: Substituting this function into the di erence equation and using (5.1.2b) yields 0 (1) ; (1) = 0 which is the consistency condition (5.6.7d). Thus, both consistency conditions within (5.6.7d) are satis ed. 41 Consistency + Stability Convergence Root Condition Figure 5.6.5: Relationship between consistency, convergence, the root condition, and stability. A summary of the relationships between the basic concepts of consistency, convergence, and stability is given in Figure 5.6.5. Thus, from left to right, stability implies that the root condition is satis ed, which together with consistency implies convergence. Reading from right to left, convergence implies that the method is consistent and satis es the root condition, which implies stability. Let us recall the de nition of A-stability introduced in Chapter 3. De nition 5.6.12. A LMM is A-stable if its region of absolute stability contains the left-half plane Re(h ) 0. Unfortunately, A-stability is very di...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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