The main purpose of the present paper is to extend and unify the relevant results by Humphries & Stuart (1994), Hill (1997) and Xiao (1996) to general linear methods and dissipative dynamic systems in Hilbert spac...The main purpose of the present paper is to extend and unify the relevant results by Humphries & Stuart (1994), Hill (1997) and Xiao (1996) to general linear methods and dissipative dynamic systems in Hilbert spaces. It is showed that, for general linear methods, dissipativity implies weak AN-stability and that, for irreducible methods, algebraic stability plus p(L(∞)) <1 implies dissipativity.展开更多
The main purpose of the present paper is to examine the existence and local uniqueness of solutions of the implicit equations arising in the application of a weakly algebraically stable general linear methods to dissi...The main purpose of the present paper is to examine the existence and local uniqueness of solutions of the implicit equations arising in the application of a weakly algebraically stable general linear methods to dissipative dynamical systems, and to extend the existing relevant results of Runge-Kutta methods by Humphries and Stuart(1994). [ABSTRACT FROM AUTHOR]展开更多
This paper deals with the delay-dependent stability of numerical methods for delay differential equations. First, a stability criterion of Runge-Kutta methods is extended to the case of general linear methods. Then, l...This paper deals with the delay-dependent stability of numerical methods for delay differential equations. First, a stability criterion of Runge-Kutta methods is extended to the case of general linear methods. Then, linear multistep methods are considered and a class of r（0）-stable methods are found. Later, some examples of r（0）-stable multistep multistage methods are given. Finally, numerical experiments are presented to confirm the theoretical results.展开更多
In this paper, the concepts of regularity and strong regularity of general. linear methods are introduced. We investigate the conditions which guarantee that general linear methods preserve asymptotic values of the sy...In this paper, the concepts of regularity and strong regularity of general. linear methods are introduced. We investigate the conditions which guarantee that general linear methods preserve asymptotic values of the systems of ordinary differential equations. This work extends the existed results of Runge-Kutta methods and linear multistep methods.展开更多
The purpose of this research is to investigate the effciency of explicit diagonally implicit multi-stage integration methods with extrapolation. The author gave detailed explanation of explicit diagonally implicit mul...The purpose of this research is to investigate the effciency of explicit diagonally implicit multi-stage integration methods with extrapolation. The author gave detailed explanation of explicit diagonally implicit multi-stage integration method and compared the base method with a technique known as extrapolation to improve the effciency. Extrapolation for symmetric Runge-Kutta method is proven to improve the accuracy since with extrapolation the solutions exhibit asymptotic error expansion, however for General linear methods, it is not known whether extrapolation can improve the effciency or not. Therefore this research focuses on the numerical experimental results of the explicit diagonally implicit multistage integration with and without extrapolation for solving some ordinary differential equations. The numerical results showed that the base method with extrapolation is more effcient than the method without extrapolation.展开更多
The extrapolation technique has been proved to be very powerful in improving the performance of the approximate methods by large time whether engineering or scientific problems that are handled on computers. In this p...The extrapolation technique has been proved to be very powerful in improving the performance of the approximate methods by large time whether engineering or scientific problems that are handled on computers. In this paper, we investigate the efficiency of extrapolation of explicit general linear methods with Inherent Runge-Kutta stability in solving the non-stiff problems. The numerical experiments are shown for Van der Pol and Brusselator test problems to determine the efficiency of the explicit general linear methods with extrapolation technique. The numerical results showed that method with extrapolation is efficient than method without extrapolation.展开更多
In this paper, for general linear methods applied to strictly dissipative initial value problem in Hilbert spaces, we prove that algebraic stability implies B-convergence, which extends and improves the existing resul...In this paper, for general linear methods applied to strictly dissipative initial value problem in Hilbert spaces, we prove that algebraic stability implies B-convergence, which extends and improves the existing results on Runge-Kutta methods. Specializing our results for the case of multi-step Runge-Kutta methods, a series of B-convergence results are obtained.展开更多
In 1992, Cooper [2] has presented some new stability concepts for Runge-Kutta methods whichis based on two slightly different test problems, and obtained the algebraic conditions that guarantee newstability properties...In 1992, Cooper [2] has presented some new stability concepts for Runge-Kutta methods whichis based on two slightly different test problems, and obtained the algebraic conditions that guarantee newstability properties. In this paper, we extend these results to general linear methods and to more generalproblem class Kστ. The concepts of (k, p, q)-secondary stability and (k, p. q)-secondary stability are introduced, and the criteria of secondary algebraic stability are also established. The criteria relax algebraicstability conditions while retaining the virtues of a nonlinear test problem.展开更多
Focuses on a study which presented some invariants and conservation laws of general linear methods applied to differential equation systems. Information on the quadratic invariants; Conservation of symplectic structur...Focuses on a study which presented some invariants and conservation laws of general linear methods applied to differential equation systems. Information on the quadratic invariants; Conservation of symplectic structure; Details on the multiple Runge-Kutta methods; Equations of the one-leg methods.展开更多
Some new concepts of stability are introduced for general linear methods, and algebraic conditions for stability of the methods are proposed which are suitable not only for implicit methods but also for explicit metho...Some new concepts of stability are introduced for general linear methods, and algebraic conditions for stability of the methods are proposed which are suitable not only for implicit methods but also for explicit methods. Our results characterize the interrelation between linear and nonlinear stability so that new evidence for the construction of efficient and stable methods is offered.展开更多
文摘The main purpose of the present paper is to extend and unify the relevant results by Humphries & Stuart (1994), Hill (1997) and Xiao (1996) to general linear methods and dissipative dynamic systems in Hilbert spaces. It is showed that, for general linear methods, dissipativity implies weak AN-stability and that, for irreducible methods, algebraic stability plus p(L(∞)) <1 implies dissipativity.
基金a grant !(No. 19871070) from NSF of China a grant!(No. A757D9I0) from Academy of Mathematics and System Sciences, Academy o
文摘The main purpose of the present paper is to examine the existence and local uniqueness of solutions of the implicit equations arising in the application of a weakly algebraically stable general linear methods to dissipative dynamical systems, and to extend the existing relevant results of Runge-Kutta methods by Humphries and Stuart(1994). [ABSTRACT FROM AUTHOR]
基金supported by National Natural Science Foundation of China(Grant No.10671078)the Program for New Century Excellent Talents in University,the State Education Ministry of China. supported in part by E-Institutes of Shanghai Municipal Education Commission (No.E03004)+3 种基金National Natural Science Foundation of China(No.10671130)Shanghai Science and Technology Commission(No.06JC14092)Shuguang Project of Shanghai Municipal Education Commission(No.06SG45)the Shanghai Leading Academic Discipline Project(No.S30405)
文摘This paper deals with the delay-dependent stability of numerical methods for delay differential equations. First, a stability criterion of Runge-Kutta methods is extended to the case of general linear methods. Then, linear multistep methods are considered and a class of r（0）-stable methods are found. Later, some examples of r（0）-stable multistep multistage methods are given. Finally, numerical experiments are presented to confirm the theoretical results.
文摘In this paper, the concepts of regularity and strong regularity of general. linear methods are introduced. We investigate the conditions which guarantee that general linear methods preserve asymptotic values of the systems of ordinary differential equations. This work extends the existed results of Runge-Kutta methods and linear multistep methods.
文摘The purpose of this research is to investigate the effciency of explicit diagonally implicit multi-stage integration methods with extrapolation. The author gave detailed explanation of explicit diagonally implicit multi-stage integration method and compared the base method with a technique known as extrapolation to improve the effciency. Extrapolation for symmetric Runge-Kutta method is proven to improve the accuracy since with extrapolation the solutions exhibit asymptotic error expansion, however for General linear methods, it is not known whether extrapolation can improve the effciency or not. Therefore this research focuses on the numerical experimental results of the explicit diagonally implicit multistage integration with and without extrapolation for solving some ordinary differential equations. The numerical results showed that the base method with extrapolation is more effcient than the method without extrapolation.
文摘The extrapolation technique has been proved to be very powerful in improving the performance of the approximate methods by large time whether engineering or scientific problems that are handled on computers. In this paper, we investigate the efficiency of extrapolation of explicit general linear methods with Inherent Runge-Kutta stability in solving the non-stiff problems. The numerical experiments are shown for Van der Pol and Brusselator test problems to determine the efficiency of the explicit general linear methods with extrapolation technique. The numerical results showed that method with extrapolation is efficient than method without extrapolation.
基金The project supported by the National Natural Science Foundation of China
文摘In this paper, for general linear methods applied to strictly dissipative initial value problem in Hilbert spaces, we prove that algebraic stability implies B-convergence, which extends and improves the existing results on Runge-Kutta methods. Specializing our results for the case of multi-step Runge-Kutta methods, a series of B-convergence results are obtained.
文摘In 1992, Cooper [2] has presented some new stability concepts for Runge-Kutta methods whichis based on two slightly different test problems, and obtained the algebraic conditions that guarantee newstability properties. In this paper, we extend these results to general linear methods and to more generalproblem class Kστ. The concepts of (k, p, q)-secondary stability and (k, p. q)-secondary stability are introduced, and the criteria of secondary algebraic stability are also established. The criteria relax algebraicstability conditions while retaining the virtues of a nonlinear test problem.
基金NSF of China(No. 19871070), Wang Kuancheng Foundation for Rewarding thePostdoctors of Chinese Academy of Sciences and the Post
文摘Focuses on a study which presented some invariants and conservation laws of general linear methods applied to differential equation systems. Information on the quadratic invariants; Conservation of symplectic structure; Details on the multiple Runge-Kutta methods; Equations of the one-leg methods.
文摘Some new concepts of stability are introduced for general linear methods, and algebraic conditions for stability of the methods are proposed which are suitable not only for implicit methods but also for explicit methods. Our results characterize the interrelation between linear and nonlinear stability so that new evidence for the construction of efficient and stable methods is offered.
