Algebraic Stability And Quantitative Error Analysis Of Two Classes Of Two-Step Runge-Kutta Methods

Stiff problems arise widely in many scientific and engineering fields such as control systems,biology,electronic network,physics and chemistry, etc. In recent more than twenty years, the study on the theory of computational methods for stiff problems developed rapidly,and we have a relative mature theory—B-theory. But this theory only fits stiff differential equations with moderate-size one-sided Lipschitz constants. Singular perturbation problems are typical classes of stiff problems. But they can't be covered by the known classical convergence theory and B-theory, due to their classical Lipschitz constants and one-sided Lipschitz constants are in general of size O (ε^-1 )(0 <ε<<1). Therefore, it's necessary to do special study about stability properties and convergence properties of their numerical methods.Rosenbrock methods and two-step Runge - Kutta methods are two classes of numerical methods in common use. The Rosenbrock methods (including two-step W-methods) can be easily implemented for solving stiff differential equations. So, its computation amount is the same as BDF methods', and less than general implicit Runge - Kutta methods'with same stages. Two-step Runge - Kutta methods have favourable stability and achieve higher order than the general one-step Runge - Kuttamethods with same stages.This paper is composed of two parts. In the first part, we study the convergence of parallel two-step W-methods and parallel partitioned two-step W-methods for two-parameter singular perturbation problems, and obtain the local and global error estimates in the variable stepsizes environment. In the second part, we study the algebraic stability of a class of the two-step Runge - Kutta methods. We also present several classes of high order two-step Runge - Kutta methods with algebraic stability. These extend the discussion of the stability of two-step Runge - Kutta methods from the linear stability to the nonlinear stability. In addition, the corresponding theoretical results are confirmed by numerical example in each chapter.