Quantitative Error Analysis Of Rosenbrock Methods For Some Classes Of Stiff Differential Equations

Because of variety and complexity of stiff problems in science and technology, the different classes of stiff problems have different structural characteristics and the essential problem-characterizing parameters. The known B-theory principally fits stiff differential equations with moderate-size one-sided Lipschitz constants. But it can't cover some typical and important classes of stiff problems in chemical kinetics, automatics control, electronic systems, mechanics etc, for example, singular perturbation problems etc. Therefore, it is important and meaning that studying error behaviors of numerical methods for stiff problems which can't be covered by B-theory. Rosenbrock methods are one class of important numerical methods in common use for solving stiff problems. The methods have almost the same computation amount as BDF methods, and less computation amount than implicit Runge-Kutta methods. These cause people's wide interest in the methods. The shortage of B-stability brings some substantive difficulties to their quantitative convergence analysis. In this paper, we study quantitative error behaviors of Rosenbrock methods for multiply-stiff singular perturbation problems, and obtain the local and global error analysis results. These extend and develop the corresponding results given by Strehmel et.al. in 1991 for singly-stiff singular perturbation problems. In this paper, we also study quantitative error behaviors of Rosenbrock methods for one class of stiff semilinear problems with stiffness contained in the variable coefficients linear part, and obtain the local and global error analysis results. These extend and develop the corresponding results given by Strehmel et.al. in 1991 for stiff semilinear problems with stiffness contained in the constant coefficients linear part.