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1、华中科技大学 硕士学位论文 三角指数拟合Runge-Kutta方法 姓名：金永虎 申请学位级别：硕士 专业：计算数学 指导教师：黄乘明 2011-05-20 I ? ? ? Runge-Kutta ?FRK ? ? Runge-Kutta ? ETFRK ? ? ? ? FRK ? ? ETFRK ? ETFRK ? FRK ? ? ETFRK ? ETFRK ? FRK ? Runge-Kutta ? ? Runge-Kutta ? ? ? ETFRK ? ? ? ? ETFRK ? ? ? ? FRK ? ? ? Runge-Kutta ?ETFRK ? ? II Abstract Bas
2、ed on the theory of functional fitting Runge-Kutta methods(FRK methods), we construct a class of methods called trigonometric-exponential fitting Runge-Kutta methods by selecting a group of new basic functions, and note them ETFRK methods. The paper begins with introducing the structure and characte
3、ristics of FRK methods, raising the concept and requirements of the basic functions. On this basis, we develop ETFRK methods which are based on a group of new basis functions. The coefficients of ETFRK methods are derived, and the characteristics of FRK methods are expanded in. After introducing the
4、 constructure theory of ETFRK methods, we give several kinds of explicit and implicit ETFRK methods. In explicit methods, we introduce extended Runge-Kutta methods for optimization, because general explicit Runge-Kutta methods cannot satisfy the requirements of the exact solution. But explicit metho
5、ds can only have two basis functions, which reduce the complexity. We define the corresponding algebraic methods of the ETFRK methods to investigate the order of the methods, which is verified in the practical experiments. In the implicit methods, we develop the methods fully in accordance with the
6、theory, so that the basis functions are rich, and we can also use the collocation techniques to control the order of the ETFRK methods, which is verified in the practical experiments. At last, we also suppose the development direction of FRK methods. Firstly, new methods can be developed. Secondly,
7、better order is expected. In addition, we can discuss the stability of the methods and so on. Key Words: Runge-Kutta methods, FRK methods, exponential fitting, trigonometric fitting, ETFRK methods ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?三三三?三三三三三三?三 ?三 ?三 三 ?三三三三三三三三三?三 ? ? ? ? ? ? ? ? ? 1 1 ? ? ? ? ? ?Betti
8、s1? Gautschi2? ? Runge-Kutta ? ? 00 ( , ) , () yf x y y xy = = ?1.1? ? Runge-Kutta ? FRK ? ?Ozawa ?3? FRK ? ?FRK ?FRK ? FRK ? ? ? FRK ? 1 k ii u = ? ? ?FRK? ?Hoang?Sidje?Cong4?Ozawa?FRK? ? ?FRK?Runge-Kutta ? ?Ozawa?x? ?Sidje?Hoang?Cong?5 ?FRK? ?FRK?x? ?Sidje?Hoang?6? ?H? 1 1 k ii Hspanu = =?FRK? 2 ?
9、HH? 1 1 k ii Hspanu = =? ? ? 1 k ii u = ? ? FRK ? ? FRK ? FRK ? ? 1 ( ) , kk ii u xxx = =? Runge-Kutta ? ? ? ? ? ? 2 1 ( )sin(),cos(),sin(),cos() r ii u xxxr xr x = =? ? ? ? 21 1 ( )sin(),cos(),sin(),cos() r ii u xxxr xr xx + = =? ? ? ?R? ? ? sine-cosine ?Sidje?Hoang ? Cong5? ?Calvo et al7? ? ? 2 11
10、 ( )exp(),exp() rjjr iijjj u xxx xx = =? ?R? ? exponetial ? ? Simos8?Vanden Berghe et al 9-10? Vanden Berghe?De Meyer ? Van Daele11? ? ? Vigo-Aguiar? Ramos12? Simos? Vigo-Aguiar13? Van de Vyver14? Calvo et al15? ? ? ? Vanden Berghe? Ixaru ? Van Daele11? ? 2 1 ( )exp(),exp(), , kk ii u xxx xxR = =? Coleman? Duxbury 16 ? 1 ( )sin(), k ii u xx = = 2 cos(), , k x xxR ? ? FRK ?A, b, c?Ozawa ?3? c ? ? ?
