tailieunhanh - Numerical Methods for Ordinary Dierential Equations Episode 11

Tham khảo tài liệu 'numerical methods for ordinary dierential equations episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 334 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS also the accumulated effect of errors generated in previous steps. We present a simplified discussion of this phenomenon in this subsection and discuss the limitations of this discussion in Subsection 421. Suppose a sequence of approximations yi y x1 y-2 y x2 . . . yn-1 y xn-i has been computed and we are now computing step n. If for the moment we ignore errors in previous steps the value of yn can be evaluated using a Taylor expansion where for implicit methods we need to take account of the fact that f yn is also being calculated. We have y xn - yn - hpo f y xn - f yn y xn - aiy xn-i - h piy xn-1 i 1 i 0 which is equal to Cp ihp 1y p 1 xn O hp 2 . In this informal discussion we not only ignore the term O hp 2 but also treat the value of hp 1y p 1 xn-i as constant. This is justified in a local sense. That is if we confine ourselves to a finite sequence of steps preceding step n then the variation in values of this quantity will also be O hp 2 and we ignore such quantities. Furthermore if y xn - yn - hpo f y xn - f yn Cp 1hp 1 y p 1 xn then the assumption that f satisfies a Lipschitz condition will imply that y xn - yn Cp 1hp 1y p 1 xn and that h f y xn - f yn O hp 2 . With the contributions of terms of this type thrown into the O hp 2 category and hence capable of being ignored from the calculation we can write a difference equation for the error in step n which will be written as Ễn y xn - yn in the form en - V aie i Khp 1 n è LINEAR MULTISTEP METHODS 335 where K is a representative value of Cp 1y p 1 . For a stable consistent method the solution of this equation takes the form en a 1 1hp 1nK 2 hi n 420a i 1 where the coefficients ni i 1 2 . k depend on initial values and xi i 1 2 . k are the solutions to the polynomial equation a A 1 0. The factor a 1 1 that occurs in 420a can be written in a variety of forms and we have az 1 pz 1 1 ơ 1 ai 2a2 kak. The value of Ca 1 1 is known as the error constant for .

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