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Numerical Methods for Ordinary Dierential Equations Episode 6

Tham khảo tài liệu 'numerical methods for ordinary dierential equations episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | RUNGE-KUTTA METHODS 159 The resulting sum is the value of t . A similar formula for i t where i is not a member of V is found by replacing 312e by aij n aki 312f k iy_E and summing this as for t . Note that although c does explicitly appear in Definition 312A or Lemma 312B it is usually convenient to carry out the summations 52s 1 akl to yield a result ck if l denotes a leaf terminal vertex of V. This is possible because l occurs only once in 312e and 312f . We illustrate the relationship between the trees and the corresponding elementary weights in Table 312 I . For each of the four trees we write t in the form given directly by Lemma 312B and also with the summation over leaves explicitly carried out. Finally we present in Table 312 II the elementary weights up to order 5. 313 The Taylor expansion of the approximate solution We show that the result output by a Runge-Kutta methods is exactly the same as 311d except that the factor y 1 is replaced by t . We first establish a preliminary result. Lemma 313A Let k 1 2 . . . . If Yi yo V - - i t hr t F t y0 O hk 313a . t r t k-1 then hf Yi - iD t hr i F t yo O hk 1 . 313b r k ơ t Proof. Use Lemma 310B. The coefficient of ơ t -1F t y0 hr t in hf Yi is n 1 i tj where t tit2 tk . We are now in a position to derive the formal Taylor expansion for the computed solution. The proof we give for this result is for a general Runge-Kutta method that may be implicit. In the case of an explicit method the iterations used in the proof can be replaced by a sequence of expansions for Y1 for hf Y _ for Y2 for hf Y2 and so on until we reach Ys hf Ys and finally y1 . 160 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Theorem 313B The Taylor expansions for the stages stage derivatives and output result for a Runge-Kutta method are Yi yo V -1- i t hr t F t yo O hn 1 i 1 2 . s 313c t r t n hf Yi E t YD t hr t F t yo O hn 1 i 1 2 . s 313d yi yo E h F t yo O hn 1 . 313e Proof. In a preliminary part of the proof we consider the sequence