Construction And Research Of The High-order Runge-Kutta Method

Ordinary Differential Equations(ODEs)are important to many fields of Natural Sciences,such as Automatic Controls?designs of Electronic Device?Trajectory calculations?the flight stabilities of Aircrafts and Missiles and studies on the stability of chemical reaction processes,which can be transformed into solving or studying of the ODEs.At present,the development of computer technology provides a powerful tool for the practical and theoretical researches of ODEs.However,there are only some special types of initial value problems of ODEs,which can be expressed analytically.It is usually difficult to get the exact solutions for the initial value problems of ODEs,and some initial value problems of ODEs even can not use exact formulas to represent,so we should rely on numerical methods to obtain numerical solutions of the initial value problems.Runge-Kutta methods are classical methods to solve the initial value problems of ODEs.In this thesis,we consider to construct high-order implicit and symplectic Runge-Kutta meth-ods(ISRK)and explicit and symmetric Runge-Kutta methods(ESRK)to solve the initial value problems of ODEsIn the third chapter,we mainly study the construction of high-order ISRK methods.To con?struct the high-order and symmetric ISRK methods,we use the techniques of W-transformation introduced by professor Hairer and Wanner.By selecting different values of the parameters ?,?and ? in the Transformation Matrix X:we can get the known methods,such as the Gauss methods,Lobatto ?A,Lobatto ?B,Lobatto?C,Lobatto ?E,Lobatto ?S,and we also obtain a new class of symmetric ISRK methods(called Lobatto ?SX).The properties of Lobatto ?SX methods we constructed can be proved directly by the results obtained by professor Hairer and Wanner.In the fourth chapter,we mainly study the construction of 6th order ESRK methods with 8 stages.We can see the difficulties and complexities of constructing high-order ESRK methods from the table(2-1).In order to overcome the difficulties of the existed methods which are used to construct high-order explicit Runge-Kutta methods,we consider the general Runge-Kutta methods of 8 stages,the general adjoint methods(in this paper,we call the symmetric adjoin-t methods)and symplectic adjoint methods(new defined),and give a new way to construct 6th order ESRK methods with 8 stages.Firstly,we give the simplification of the simplifying assump-tions B(p)(p<6),C(1)and D(1)by using the explicit Runge-Kutta method of 8 stages(4.2.1)and symplectic and symmetry method(4.2.5)(see table 4-2)obtained by the symplectic adjoint method(4.2.4),under the conditions of A*= As*and the symmetry of the quadrature formula(b,c).Then,we prove that the simplifying assumptions Cs*(i)and Ds*(i)(i = 1,2)of the sym-plectic and symmetry method(4.2.5)are equivalent.Secondly,by using the relevant conclusions of A*As*?C(1)(?)D(1)?CS*(1)(?)DS*(1)?CS*(2)(?)DS*(2)and the symmetry of the quadrature formula(b,c),we simplify the 33 order conditions which should be satisfied by an 8 stages 6th order ESRK method in three stages.Through the above simplification processes,we obtain that the 6th order ESRK methods of 8 stages should satisfy the order condition(4.2.55).Thirdly,under the condition of regarding c2,c3,c4,b2,b3,a43(or a42)as free variables,we get the expressions for the coefficients a32,a42(Or a43),a52,a53,a54,a62,a63,a72 and b4 of the 8 stages 6th order ESRK method's Butcher Table.Finally,we construct a class of 6th order ESRK methods of 8 stages by selecting free variables of a43(or a42),c2 and c4 and using formulas of(4.2.6),(4.2.11),(4.2.13),(4.2.14),(4.3.7),(4.3.8),(4.3.9),(4.3.10),(4.3.11),(4.3.12),(4.3,13),(4.3.14)and(4.3.20).