Tham khảo tài liệu 'numerical methods for ordinary dierential equations episode 10', 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 299 where the result is interpreted as meaning that E t 1 t -1 Tk t k i for any t G T. Since E takes the exact solution to a differential equation through one unit step h it is natural to ask how we would represent the solution at a general point Oh advanced from the initial point. We write this as E 0 and we note that E ỡ t Or t E t for all t G T. We can generalize 387d in the form 00 k E 1 s u Tk and note that for O an integer n we have E n En. This property is to some extent characteristic of E and we have Theorem 387A If a G G1 such that a T 1 and m is an integer with m G 0 1 1 then a. m am implies that a E. Proof. For any tree t T we have a m t r t ma t Q1 and am t ma t Q2 where Q1 and Q2 are expressions involving a u for r u r t . Suppose that a u has been proved equal to E u for all such trees. Then a m t r t ma t Q1 am t ma t Q2 E m t r t mE t Q1 E m t mE t Q2 so that a m t am t implies that r t m m a t E t 0 implying that a t E t because r t m m whenever r t 1 and m G 0 1 1 . Of the three excluded values of m in Theorem 387A only m 1 is interesting. Methods for which a -1 a 1 have a special property which makes them of potential value as the source of efficient extrapolation 300 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS procedures. Consider the solution of an initial value problem over an interval xq x using n steps of a Runge-Kutta method with stepsize h x xQ n. Suppose the computed solution can be expanded in an asymptotic series in h y x Ch. 387e i 1 If the elementary weight function for the method is a then the method corresponding to a 1 1 exactly undoes the work of the method but with h reversed. This means that the asymptotic error expansion for this reversed method would correspond to changing the sign of h in 387e . If a a 1 1 this would give exactly the same expansion so that 387e is an even function. It then becomes possible to extend the applicability of the method by extrapolation in even powers only. 388 Some .
