变延迟微分方程隐式Euler法的收缩性_英文_

变延迟微分方程隐式Euler法的收缩性_英文_ Ξf or Delay Differential Equations 1 2YU Yue xin , L I S hou f u ()Department of Mathematics , Xiangtan University , Xiangtan 411105 China () 【Abstract】: This paper discusses the nonlinear stability of implicit Euler method for delay differential equationsDDEs. We establish a sufficient condition for implicit Euler method to be contractive when applied to any given DDEs with a variable delay , where we only require the delay term being a nonnegative continuous function. Key words : delay differential equations , implicit Euler method , contractivity AMS Subject Classif ication : 65L06 变延迟微分方程隐式 Euler 法的收缩性 余越昕 , 李寿佛 ( )湘潭大学数学系 ,湖南 湘潭 411105 ( ) τ【摘 要】 研究隐式 Euler 法关于变延迟微分方程的收缩性 ,在对延迟量t的变化不作任何实质性限制的条件下 ,获得了 方法收缩的充分条件. 关键词 :变延迟微分方程 ; 隐式 Euler 法 ; 收缩性 () 中图分类号 :O241 . 8 文献标识码 :A 文章编号 :1000 5900 200402 0112 04 0 Introduction In 1989 , Torelli 7 discussed the nonlinear stability of numerical methods for DDEs. He introduced the con 2 cepts of RN - and GRN - stability and proved that implicit Euler method is GRN - stable . Thereafter , the nonlin2 ear stability of numerical methods have been further studied , and a great number of results have already been ob2 () tained cf . 1 ,4 - 6 ,10 - 11 . However , the above studies all aimed at the case that the delay term was a posi2 tive constant . To our knowledge , so far only the papers of Zennaro 9 , Bellen , Guglielmi and Zennaro 3 , Wang Wenqiang and Li Shoufu 8 and Bellen 2 were concerned with DDEs with variable delays. However , all the pa 2 pers required that the variable delays satisfied some severe additional conditions except the last one which treated di2 N agonally split Runge - Kutta methods applied to DDEs in a space Cwith maximum norm. In the present paper , we discuss the nonlinear stability of implicit Euler method for DDEs. We establish a sufficient condition for implicit Eu2 ler method to be contractive when applied to any given DDEs with a variable delay , where we only require the delay term being a nonnegative continuous function. In section 1 , we present the main result and prove it to be true . In section 2 , some numerical tests are given that confirm the theoretical result . 1 Nonlinear sta bility of implicit Euler method f or DD Es N Let〈. , . 〉be an inner product on Cand ‖. ‖the corresponding norm , consider the following initial value problems of DDEs. ( ) ( ( ) ( τ( ) ) ) y′t= f t , y t, y t - t, t ?0 , ()1. 1 ( ) φ( ) y t = t , t ?0 and Ξ Received Date :2003 - 03 - 20. ( ) ( ( ) ( τ( ) ) ) z′t= f t , z t, z t - t, t ?0 , ()1. 2 ( ) Ψ ( ) z t = t , t ?0 N N N N τ( ) φ Ψ ) ( ) ( where t?0 t ?0,,: - ?,0 ] ?Care given continuous functions , and f : [ 0 , + ?×C×C?C is a given continuous mapping which satisfies the following conditions :2 ( ) ( ) ( ) αRe < u - u ,f t , u , v- f t , u , v> ?t‖u - u ‖,1 2 1 2 1 2 N t ?0 , u, u, v ?C , 1 2 () 1. 3 N ( ) ( ) ( ) β‖f t , u , v - f t , u , v ‖ ?t‖v - v ‖, t ?0 , u , v , v ?C,1 2 1 2 1 2 () () ( ) ( ) and we assume that the problems 1 . 1and 1 . 2have unique solutions y t and z t , respectively. ) (Apply the implicit Euler method to the initial problem 1 . 1, we have ( ) () y= y+ hf t, y, ?y , 1. 4n + 1 n n + 1 n + 1 n + 1 φ() ( ) where y= 0, yis an approximation to the exact solution y twith t= nh , and ?y denotes an approxima2 0 n n n + 1 ( τ( ) ) tion to y t- t. In this paper , we use the linear interpolation procedure to obtain ?y . n + 1 n + 1 n + 1 τ( ) ( δ) δ( Let t = m- h , with integer m?1 and ?0 ,1 ] , we definen + 1 n + 1 n + 1 n + 1 n + 1 δδ) (()y ?= y+ 1 - y, 1. 5 n + 1 n + 1 n - m + 2 n + 1 n - m + 1 n +1 n +1 φ( τ( ) ) τ( ) and ?y = t- tfor t- t?0. n + 1 n + 1 n + 1 n + 1 n + 1 () Similarly , apply the implicit method to the problem 1 . 2, we have ( ) () z= z+ hf t , z,z ?, 1. 6n + 1 n n + 1 n + 1 n + 1 with δδ) (()?z = z+ 1 - z, 1. 7 n + 1 n + 1 n - m + 2 n + 1 n - m + 1 n +1 n +1 Ψ ( τ( ) ) τ( ) and z ?= t - t for t - t ?0. n + 1 n + 1 n + 1 n + 1 n + 1 () Theorem 1 . 1 Let { y} and { z} denote two approximation sequences produced by the method 1 . 4with n n () () a fixed step size h applied to the problems 1 . 1and 1 . 2, respectively. We have ( ) ψ( ) φ‖y- Z‖?max ‖t- t‖, n ?0 ,n n () 1. 8t ?0 provided that ()1. 9 β( ) α( ) t ?- t for t > 0. () Note that the inequality 1 . 8characterizes the contractivity of implicit Euler method. ( ) ψ( ) φProof Let S = max ‖t - t ‖, w= y- z, n n n 1 ?0 ( ) ( ) Q= f t, y, y ?- f t, z,?z , n + 1 n + 1 n + 1 n + 1 n + 1 n + 1 n + 1 then w= w+ hQ. n + 1 n n + 1 Hence2 22 2 + 2 hRe < w , Q > + h = ‖w ‖= ‖w ‖‖Q ‖n + 1 n n n + 1 n + 1 2222 ‖w ‖ ? ‖w ‖ + 2 hRe < w - hQ , Q > + h + 2 hRe < w, Q> . ‖Q ‖n + 1 n + 1 n +1 n n + 1 n + 1 n + 1 n () 1. 10 () () () It follows from 1 . 3, 1 . 9and 1 . 10that2 2 ( ) ( ) ?‖w ‖+ 2 hRe < w, f t , y, y ?- f t , z, y ?> ‖w ‖n + 1 n + 1 n + 1 n + 1 n + 1 n + 1 n + 1 n + 1 n ( ) ( ) + 2 hRe < w, f t , z, y ?- f t , z,z ?> n + 1 n + 1 n + 1 n + 1 n + 1 n + 1 n + 1 22 α( ) β( ) ?‖w‖ + 2 ht ‖w‖ + 2 ht ‖w‖?‖y ?- z ?‖ n n + 1 n + 1 n + 1 n + 1 n + 1 n + 1 2 β( ) ( ) ()ω ?‖w ‖+ 2 ht ‖w ‖‖?y - z ?‖- ‖‖. 1. 11 n n + 1 n + 1 n + 1 n + 1 n + 1 It is obvious that ()‖w‖ ?S , j ?0. 1. 12j For any given nonnegative integer n , we thus only need to prove that ‖w‖?S under the inductive assumption n + 1 that ‖w‖?S for k ?n . k In fact , if ‖w‖> S , then we have n + 1 ()‖w‖ > ‖w‖, 1. 13 n + 1 n ()and therefore together with 1 . 11 2 2β( ) ( ) ()w‖ - ‖w‖ ?2 htw‖y - z ‖- ‖w‖. 1. 14 0 < ‖??‖‖n + 1 n n + 1 n + 1 n + 1 n + 1 n + 1 () β( ) if t= 0 , then 1 . 14yields n + 1 ‖w‖ ? ‖w‖,n + 1 n () β( ) β( ) () that is in contradiction with 1 . 13, so we do have t> 0 . Since t> 0 , 1 . 14leads to n + 1 n + 1 ‖y z ?- ?‖- ‖w‖ > 0 ,n + 1 n + 1 n + 1 i . e . ) ‖+ 1 ) (δ) ( ω( δ‖‖< ‖y?- z?‖= ‖y- Z- z+ 1 - y+ 2 n - m+ 1 n - mn + 1 n + 1 n +1 n + 1 n - m+ 2 n + 1 n - mn +1 n +1 n +1 n +1 ()‖. 1.15 + 1 δδ) (?‖y‖+ 1 - ‖y- z- Zn + 1 n - m+ 2 n - m+ 2 n + 1 n - m+ 1 n - mn +1 n +1 n +1 n +1 () if m= 1 , then 1 . 15yields n + 1 δδ) (‖w‖ < ‖w‖ + 1 - ‖w‖, n + 1 n +1 n +1 n + 1 n hence ‖w, ‖ < ‖w‖n + 1 n () () that is in contradiction with 1 . 13. Therefore , we have m> 1 , and 1 . 15thus yields n + 1 δδ) (‖w‖ < S + 1 - S = S . n + 1 n + 1 n + 1 This is in contradiction with the assumption ‖w‖> S , and completes the proof of the theorem. n + 1 2 Numerical test Consider the following initial value problems of DDEs ( ) () ( ) ( ) y′t = - 2 + sin t y t + y t - 2 + co s t , t ?0 , ()t 2. 1 ( ) y t = e, t ?0 , and ( ) () ( ) ( ) z′t= - 2 + sin tz t+ z t - 2 + co s t , t ?0 , 2 t ()2. 2 ( ) z t= e, t ?0 , () () () It is easily verified that the condition 1 . 3and 1 . 9are satisfied. Applying implicit Euler method 1 . 4 () () with step size h > 0 to the problems 2 . 1and 2 . 2, we find the numerical solutions { y} and {z} respectively , n n which together with the difference y- zare listed in the table 2 . 1 , 2 . 2 and 2 . 3 for different step size h . n n Ta b. 2. 1 h = 0. 5 , t y z y - z n n n n n 1 0 . 364 363 0 . 544 283 - 0. 179 920 5 0 . 325 499 0 . 343 627 - 0. 018 127 50 0 . 000 095 0 . 000 095 0 . 000 000 Ta b. 2. 2 h = 0. 1 t y z y - z n n n n n 1 0 . 302 958 0 . 498 463 - 0. 195 505 5 0 . 275 796 0 . 297 164 - 0. 021 368 50 0 . 000 042 0 . 000 042 0 . 000 000 Ta b. 2. 3 h = 0. 01 t y z y - z n n n n n 1 0 . 287 256 0 . 486 805 - 0. 199 548 5 0 . 268 134 0 . 289 467 - 0. 021 333 20 0 . 007 689 0 . 007 689 0 . 000 000 It is seen from the tables that the implicit Euler method is numerical stable , which confirm our theoretical re2 sult . 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Stability of numerical methods for delay differential equationsJ . J Comput Appl Math , 1989 ,25 :15 - 26 . 8 Wang Wenqiang , Li Shoufu. The numerical stability of one - leg methods for nonlinear delay differential equations with a variable delayJ . Numer Math Sinica , 2002 ,24 : 321 - 335 . 9 Zennaro M. Asymptotic stability analysis of Runge - Kutta methods for nonlinear systems of delay differential equationsJ . Numer Math , 1997 , 77 : 549 - 563 . 10 Zhang Chengjian , Zhou Shuzi . Nonlinear stability and D - convergence of Runge - Kutta methods for delay differential equationsJ . J Comput Ap 2 pl Math , 1997 ,85 : 225 - 237 . 11 Zhang Chengjian. Stability and D - convergence of numerical methods for functional differential equationsD . 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