Stability And Convergence Properties Of Multistep Runge-Kutta Methods For The Stiff Semilinear Systems

The stiff semilinear problems are widely appears in the practical applicationsand scientific and engineering computation. For example, some parabolic initial boundary-value problem, after simi-discrete on the spatial variable, can be regarded as a stiff similinear ordinary differential initial-value problem.We consider the stiff semilinear problemswhere, the stiffness of the problems contained in the variable coefficient linear part J(t)y(t). In this paper we study the stability and convergence properties of multistep Runge-Kutta methods. Under the suitable assumptions on the matrix J(t), the results on the unique solvability of the stage equations and the stability and B-convergence of a class of multistep Runge-Kutta methods are obtained respectively. Because of the methods need not to satisfy the algebraic stability, in comparison with the commonly multistep Runge-Kutta method, our methods are more generaly and wildly. At the same time, the results in present paper can be regarded as the promotion and the development of the corresponding results which were obtained by Burrage, Hudsdorfer and Verwer, Calvo. GonzalezPintoand Montijano, etc., from the semilinear problems with the Runge-Kutta methods.