tailieunhanh - Numerical Methods for Ordinary Dierential Equations Episode 4

Tham khảo tài liệu 'numerical methods for ordinary dierential equations episode 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 88 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Table 222 I Errors in the numerical solution of the orbital problem 201d with zero eccentricity through a half period using 222a n yi error Ratio y2 error Ratio 32 64 128 256 512 1024 n y3 error Ratio y4 error Ratio 32 64 128 256 512 1024 experiments we report here the first step is taken using the Runge-Kutta method introduced in the previous subsection. The errors are shown in Table 222 I and we see that for this problem at least the results are just as good as for the Runge-Kutta method 221a and 221b even though only one derivative is computed in each step. In fact for components 1 and 4 better than second order convergence is observed. 223 Use of higher derivatives For many practical problems it is possible to derive formulae for the second and higher derivatives of y making use of the formula for y given by a differential equation. This opens up many computational options which can be used to enhance the performance of multistage Runge-Kutta and multivalue multistep methods. If these higher derivatives are available then the most popular option is to use them to evaluate a number of terms in Taylor s theorem. Even though we consider this idea further in Section 25 we present a simple illustrative example here. Consider the initial value problem y yx y y 0 2 223a NUMERICAL DIFFERENTIAL EQUATION METHODS 89 Figure 223 i Errors in problem 223a using Taylor series with orders p 1 2 3 4 with solution y x XXP x . 2 fX exp 2x2 dx By differentiating 223a once twice and a third time it is found that y x 2y