Numerical Analysis For Rosenbrock-RK Implicit-explicit Methods For Nonlinear Stiff Systems

In some actual applications (such as the chemical reaction and atmospheric chemistry) and the spatial discretization of the initial-boundary value problems of partial differential equations, we often get some large stiff systems, whose parts present the different levels of stiffness. To reduce the amount of calculation and get the relatively accurate numerical solutions of these problems, we use the Rosenbrock methods for stiff parts and the explicit Runge-Kutta methods for nonstiff parts.In this paper, we study the consistency, stability and convergence of the Rosenbrock-RK IMEX methods for two kinds of stiff initial value problems. The full text is composed of five chapters.The first chapter firstly introduces the related research background, trends, and the existing results, and it presents the discussed two classes of stiff problems in this paper.The second chapter discusses the order conditions of the Rosenbrock-RK IMEX methods, Based on the tree theory, we obtain the third-order conditions, and the expression of the higher-order conditions is deduced in theory.The third chapter analyzes the stability of the Rosenbrock-RK IMEX methods, and gets some L-stable numerical methods with the second order and the third order.The fourth chapter and the fifth chapter obtain the error-analysis results of the Rosenbrock-RK IMEX methods for the problem classes Ⅰ and Ⅱ and the correspond-ing numerical experiments are given to verify the obtained results.