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13 yn are presented in table 211 for 2 0 10 the error

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Unformatted text preview: .1.2 display the solution and global error at = 0 1 for = ;100 and several choices of . It appears that n is decreasing linearly with . Let us verify that the results of Example 2.1.1 hold more generally. We will need de nitions of the various errors prior to establishing a result. n t h e : h De nition 2.1.1. The local error is the amount by which the numerical and exact solu- tions di er after one step assuming that both were exact at and prior to the beginning of the step. 5 tn n 0 1 2 3 4 5 6 7 8 9 10 yn 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 en 1.00000 0.90000 0.81000 0.72900 0.65610 0.59049 0.53144 0.47830 0.43046 0.38742 0.34868 0.00000 0.00484 0.00873 0.01182 0.01422 0.01604 0.01737 0.01829 0.01886 0.01915 0.01920 Table 2.1.1: The solution and error of = , (0) = 1, obtained by Euler's method with = ;100 and = 0 001. T = n n 102 10 3 2 66 10 5 4 15 10 1 103 10 4 4 32 10 5 4 91 10 2 104 10 5 4 52 10 5 4 99 10 3 y h 0 y y : n h T =n y ; e =e ; : ; ; : ; ; : ; : ; : ; : Table 2.1.2: The solution and error of = , (0) = 1, at = 0 1 obtained by Euler's method with = ;100 and = 10 3, 10 4, 10 5. y 0 ; h y y ; T : ; Example 2.1.2. According to De nition 2.1.1, the local error dn at tn is dn assuming that yk = ( n) ; yt = ( k ), = ; 1 ; 2 yt k dn n n yt yt ( ) = ( n 1) + yt ; hy 0 ( )) (2.1.4b) : tn;1 ( n 1) + t ( hf tn;1 y tn;1 ; Expanding ( n) in a Taylor's series about y tn 0. For Euler's method (2.1.2), ::: = ( n) ; ( n 1) ; yt (2.1.4a) yn 2 h ; 2 y 00 ( n) tn;1 < n < tn : Using (2.1.1) to eliminate ( n 1 ) and substituting the result into (2.1.4b) yields y 0 dn t ; = 2 h 2 y 00 ( n) tn;1 < n < tn : The local error for Euler's method is illustrated in Figure 2.1.1. 6 (2.1.4c) The local discretization error or local trunctaion error is closely related to the local error. Before de ning it, let us write the di erence equation (2.1.2) in a form that more closely resembles the ODE. Thus, we de ne the di erence operator ( )) (2.1.5a) L ( ) := ( n) ; ( n 1) ; ( h u tn ut ut ; f tn;1 u tn;1 h : De nition 2.1.2. The local discretization error or local truncation error is the amount by which the exact solution of the ODE fails to satisfy the di erence operator. The local discretization error for Euler's method (2.1.2) is n = Lh ( n) yt (2.1.5b) : Comparing this result with (2.1.4b), reveals that n = n = ( 2) ( n) however, we shall see that the local and local discretization errors can have a more complex relationship for other di erence methods. d =h h= y 00 De nition 2.1.3. A di erence method is consistent to order if n = ( p). If p Oh p 1, the di erence scheme is said to be consistent. Thus, Euler's method (2.1.2) is consistent of order one or, simply, consistent. A numerical method converges if its global discretization error (2.1.3) approaches zero as the mesh is re ned. De nition 2.1.4. Consider a calculation performed on 0 n =1 2 ::: N <t T with h = T =N and . A numerical method is convergent if lim N !1 h!0 Nh=T j nj = 0 e 8 20 ] n N: (2.1.6) If en = O(hp), the method i...
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