GAUSS格式的拟渐近性(英文0714).doc

ON SPURIOUS ASYMPTOTIC NUMERICAL SOLUTIONS OF GAUSS SCHEME Liu Wenhai Fujian International Business and Economic College Abstract In this paper, we discuss characteristics of Gauss scheme for numerical ordinary differential equations, which is a one-step high-order accurate difference scheme. We apply such scheme to some equations, and analyze the fixed point and stability accordingly. Key words: Ordinary differential equations，difference scheme，stability ，fixed point Introduction For most of the ordinary differential equations, their analytical expressions are not easily identified. Occasionally, even if the closed form of solutions can be found, these solutions turn out to be impractical because of the large amount of calculation involved. Under such circumstances numerical methods are introduced to solve ordinary differential equations. With regard to the numerical scheme of ordinary differential equations, H.C. Yee, P.K. Sweby, Some Aspects of Numerical Uncertainties in Time Marching to Steady-State Computations, AIAA-96- 2052, 27th AIAA Fluid Dynamics Conference, June 18-20, 1996, New Orleans, LA, AIAA J., 36 (1998), 712-724.the solution of different scheme may become periodic, chaotic, and divergent once the step length exceeds its stable boundary. For example, using an explicit Euler difference scheme to solve the following ordinary differential equations（.）We obtain different equations（.）Here we let denote the time step of different scheme , . If 2 and (1.2) is convergent, the solutions to (1.2) are convergent to 1, which is the fixed point of the ordinary differential equation (1.1). If .43, the solutions of (1.2) possess double periodicity; If 2.43 3, the solutions of (1.2) have multiple periodicity chaos phenomenon. If 3, the solutions to (1.2) dissipates as a result.This paper aims to construct the one-step explicit different scheme of ordinary differential equation (1.1), using GAUSS type integration formula. Furthermore, the different scheme obtained is applied to the ecological community growth model , B. Sjgreen, H.C. Yee, Low Dissipative High Order Numerical Simulations of Supersonic Reactive Flows, RIACS Report 01-017, NASA Ames research center, May 2001, Proceedings of the ECCOMAS Computational Fluid Dynamics Conference 2001, Swansea, Wales, UK, September 4-7, 2001. and discussion on its fixed point and stability is conducted accordingly.GAUSS difference scheme Consider the first order system (2.1) Suppose that h is the time step and We further assume the precision solution of system (2.1) to be in x=, and its approximation to be . According to (2.1), we have Using GAUSS type integration formula Trip to jianzhong Deng and zhixing Liucalculation method transportation university publisher in Xian, 2001. with third order accuracy, Where , Hence, we have the GAUSS type different scheme (2.2) Theorem 2.1 If function and are two ordered continuous differentiable, then different equation (2.2) is two ordered accurate . Trip to jianzhong Deng and zhixing Liucalculation method transportation university publisher in Xian, 2001. Theorem 2.2 If function is two ordered continuous differentiable, then the different scheme (2.2) is convergent . Applying GAUSS different scheme (2.2) to solve the ordinary differential equations (1.1), we obtain the different equation (2.3） =If =0 , then * is the fixed point of (2.3). Besides, the stability of (2.3) around fixed point * is necessitated by the inequality |1+h(*,h)|1.According to the above analysis, we have the following theorem Theorem 2.3 The fixed points of GAUSS different scheme (2.3) and its corresponding stabilized zones are as followsFor fixed point *= 1，its stabilized zone is 2；For fixed point * = , its stabilized zone is 2.Here ,*= 0，* = .Moreover, fixed point*= 0 and * = are unstabilized.Proof In（2.3）,let =0Then , By solving those fourth order quare algebraic equations , Qinyang Lee , nengchao Wang and dayi Yi Number analysis Chin Hua university publisher, 2001 we easily obtain that the fixed points of (2.3) are *= 0，*= 1，* = ，* = 。Next, we will discuss each stabilized zones of the fixed points.When y*=0, since , fixed point *= 0 is unstabilized.When* = ,So, fixed point * = is unstabilized.When *= 1, we have . To satisfy the inequality |1+h(*,h)|1 requires , that is ,0r2. Therefore, for *= 1, its stabilized zone is . When * = ，we have,That is, . Solving this inequality gives us . In other words, the stabilized zone for fixed point * = is . We know from the above that * = 1 and * = 0 are the fixed points of the system (1.1), while * = and * = are not the fixed points of system (1.1). Therefore * = and * = are quasi fixed points. 3 Numerical experimentsIn this section, we apply GAUSS of different scheme (2.2) to solve simple ecology growth model and then discuss its fixed points and stability. For convenient, we let a=1, and compute by choosing different initial values y0. The results are shown below in the diagrams. When the initial values are less than 1.0, we can discern from the above calculation. If the time step length 2.4, its solutions will quickly converge to the fixed point, If 2.42.6,its solutions will be bifurcated; H. C. Yee , M. Vinokur , M. J. Djomehri, Entropy splitting and numerical dissipation, Journal of Computational Physics, v.162 n.1, p.33-81, July 20, 2000 If 2.62.8,its solutions will appear quadfurcated or multi-furcated; If 2.83.2, its solutions will become chaotic; If 3.2, its solutions will quickly diverge