Tham khảo tài liệu 'numerical methods for ordinary dierential equations episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 264 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS where the coefficient of yn-1 is seen to be the stability function value R hL 1 hLb I - hLA -11. By rearranging this expression we see that yn R hL yyn-1 - g xn-iỸj g xn-i hb G hLb I - hLA -1 hAG - G - g xn-1 R hL yn-1 - g xn-1 g xn - Ễ0 - hLb I - hLA 1Ễ where Ễ0 h g xn-1 h d - h big xn-1 hci J0 i 1 is the non-stiff error term given approximately by 362d and 6 is the vector of errors in the individual stages with component i given by h g x n-i hfi d - h aijg x n-i hcj . Jo j i If L has a moderate size then hLb I - hLA -1e can be expanded in the form hLb I hLA h2L2A2 e and error behaviour of order p can be verified term by term. On the other hand if hL is large a more realistic idea of the error is found using the expansion I - hLA -1 - hA-A- - and we obtain an approximation to the error g xn - yn given by g xn - yn R hL g xn-1 - yn-1 Ễ0 - b A-1e - h-1L-1b A-2e - h-2L-2b A-3e------. Even though the stage order may be low the final stage may have order p. This will happen for example if the final row of A is identical to the vector b . In this special case the term b A-1e will cancel 60. In other cases the contributions from b A-1e might dominate 60 if the stage order is less than the order. Define nn Ễ0 hLb I - hLA -1e n 0 RUNGE-KUTTA METHODS 265 with no defined as the initial error g x0 y0. The accumulated truncation error after n steps is equal to n n Y R hl- ni 2R n im. i 0 i 0 There are three important cases which arise in a number of widely use methods. If R o 0 as in the Radau IA Radau IIA and Lobatto IIIC methods or for that matter in any L-stable method then we can regard the global truncation error as being just the error in the final step. Thus if the local error is O hq 1 then the global error would also be O hq 1 . On the other hand for the Gauss method with s stages R o 1 s. For the methods for which R o 1 then we can further approximate the global error as the integral of the local truncation .
