If 0 the parasitic solution 1nc2 e hn 30 decays as n

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Unformatted text preview: e starting values by a xed amount produces a bounded change in the numerical solution for all h 2 (0 ^ ) and all t 2 0 T ]. h This de nition is, once again, too general for practical veri cation. As usual, we rely on the test equation y0 = y y(0) = : (5.6.4) To conduct a stability analysis, we apply (5.6.4) to (5.1.2) to obtain k X i=0 ( iyn;i ; h i yn;i) = 0: (5.6.5) This k th order constant-coe cient di erence equation can be solved in the manner used for Example 5.6.1 thus, assume a solution of the form yn = c n (5.6.6) and substitute it into (5.6.5) to obtain k X ( ic n;i ; h Dividing by the common c i=0 n;k factor, ic n;i) = 0: we nd that is a root of the polynomial ( );h ( )=0 where ( )= k X i=0 31 i k;i (5.6.7a) (5.6.7b) and ( )= k X i i=0 k;i: (5.6.7c) Equations (5.6.7b) and (5.6.7c) are called the rst and second characteristic polynomials of the LMM, respectively. Using (5.6.3), we see that the LMM is consistent if and only if 0 (1) ; (1) = 0 (1) = 0: (5.6.7d) Equation (5.6.7a) has k solutions, i, i = 1 2 : : : k. One solution, say 1, approximates the solution of the test problem (5.6.4). The remaining k ; 1 solutions are parasitic. If the roots of (5.6.7a) are distinct then the general solution of (5.6.5) is yn = k X i=1 ci in: (5.6.8a) The assumed solution (5.6.6) must be modi ed when some of the roots are equal. Suppose, for example, that i is a root of multiplicity , then the solutions corresponding to i are (ci + ci+1 n + ci+2n2 + : : : + ci+ ;1 n ;1 ) in: (5.6.8b) Note the similarities between this analysis and that of constant-coe cient ODE 3]. Since the exact solution of the test equation (5.6.4) is y(tn) = eh n = (eh )n (5.6.9) the principal root 1 of (5.6.7) must be an approximation of eh . If the remaining roots j ij > 1, i = 2 3 : : : k, the corresponding parasitic solutions grow as n increases and the method, at the very least, cannot be absolutely stable. Let us emphasize that all roots i , i = 1 2 : : : k, of (5.6.7a-c) depend continuously on the parameter h by writing ^ i = i (h ), i = 1 2 : : : k. If j i(0)j < 1, i = 1 2 : : : k, there is a...
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