tailieunhanh - Numerical Methods for Ordinary Dierential Equations Episode 8

Tham khảo tài liệu 'numerical methods for ordinary dierential equations episode 8', 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 229 Lobatto IIIB s 0 7-ỤTĨ 14 1 2 7 V2Ĩ 14 1 5 p 8 1 -7-Un 20 120 _ 1 343 9ỤTĨ 20 2520__ 1 49 12ỤĨ2 20 360 __ 1 343 69UŨ 20 2520__ 1 119-3ỤTĨ 20__360 1 49 20 180 1 -7 UÕĨ 0 15 120 56-15UÕĨ 343-69UÕĨ 0 315 2520 8 49-12UĨ2 0 45 360 56 15UÕĨ 343-9U2Ĩ 0 315 2520 13 119 3UÕĨ 0 45 360 16 49 X 45 180 20 Lobatto IIIC s 5 p 8 0 X 7 20 60 7-U2Ĩ 1 29 14 20 180 1 1 329 105UÕĨ 2 20 2880 7 UÕĨ 1 203 30U2Ĩ 14 20 1260 1 1 49 20 180 1 49 20 180 2 7 1 15 60 20 47-15UÕĨ 203-30U2Ĩ 3 315 1260 140 73 329- 105UÕĨ 3 360 2880 160 47 15UÕĨ 29 3 315 180 140 16 49 1 45 180 20 16 49 1 45 180 20 Exercises 34 Show that there is a unique Runge-Kutta method of order 4 with s 3 for which A is lower triangular with a11 a33 0. Find the tableau for this method. Show that the implicit Runge-Kutta given by the tableau 0 0 0 0 0 1 4 1 8 1 8 0 0 7 1 14 3 0 10 100 25 20 1 2 7 0 5 7 0 1 32 250 5 14 81 567 54 has order 5. Find the tableau for the Gauss method with s 4 and p 8. Show that Gauss methods are invariant under reflection. 230 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 35 Stability of Implicit Runge-Kutta Methods 350 A-stability A a -stability and L-stability We recall that the stability function for a Runge-Kutta method 238b is the rational function R z 1 zb I - zA 11 350a and that a method is A-stable if R z 1 whenever Re z 0. For the solution of stiff problems A-stability is a desirable property and there is sometimes a preference for methods to be L-stable this means that the method is A-stable and that in addition R x 0. 350b Where A-stability is impossible or difficult to achieve a weaker property is acceptable for the solution of many problems. Definition 350A Let a denote an angle satisfying a e 0 n and let S a denote the set of points x iy in the complex plane such that x 0 and tan a x y tan a x . A Runge-Kutta method with stability function R z is A a -stable if R z 1 for all z e S a . The region S a is illustrated in Figure 350 i in