We present an algorithm for determining the stepsize in an explicit Runge-Kutta method that is suitable when solving moderately stiff differential equations. The algorithm has a geometric character, and is based on a ...We present an algorithm for determining the stepsize in an explicit Runge-Kutta method that is suitable when solving moderately stiff differential equations. The algorithm has a geometric character, and is based on a pair of semicircles that enclose the boundary of the stability region in the left half of the complex plane. The algorithm includes an error control device. We describe a vectorized form of the algorithm, and present a corresponding MATLAB code. Numerical examples for Runge-Kutta methods of third and fourth order demonstrate the properties and capabilities of the algorithm.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
The theory of B-convergence for general linear methods is extended to general nonlinear muhivalue methods and to nonlinear stiff problems in Banach spaces. Moreover, using the extended theory, a class of high-order B-...The theory of B-convergence for general linear methods is extended to general nonlinear muhivalue methods and to nonlinear stiff problems in Banach spaces. Moreover, using the extended theory, a class of high-order B-convergent multistep-multiderivative methods is constructed.展开更多
This paper presents a study on the development and implementation of a second derivative method for the solution of stiff first order initial value problems of ordinary differential equations using method of interpola...This paper presents a study on the development and implementation of a second derivative method for the solution of stiff first order initial value problems of ordinary differential equations using method of interpolation and collocation of polynomial approximate solution. The results of this paper bring some useful information. The constructed methods are A-stable up to order 8. As it is shown in the numerical examples, the new methods are superior for stiff systems.展开更多
文摘We present an algorithm for determining the stepsize in an explicit Runge-Kutta method that is suitable when solving moderately stiff differential equations. The algorithm has a geometric character, and is based on a pair of semicircles that enclose the boundary of the stability region in the left half of the complex plane. The algorithm includes an error control device. We describe a vectorized form of the algorithm, and present a corresponding MATLAB code. Numerical examples for Runge-Kutta methods of third and fourth order demonstrate the properties and capabilities of the algorithm.
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Project supported by the National Natural Science Foundation of China.
文摘The theory of B-convergence for general linear methods is extended to general nonlinear muhivalue methods and to nonlinear stiff problems in Banach spaces. Moreover, using the extended theory, a class of high-order B-convergent multistep-multiderivative methods is constructed.
文摘This paper presents a study on the development and implementation of a second derivative method for the solution of stiff first order initial value problems of ordinary differential equations using method of interpolation and collocation of polynomial approximate solution. The results of this paper bring some useful information. The constructed methods are A-stable up to order 8. As it is shown in the numerical examples, the new methods are superior for stiff systems.
