Stability and B-convergence properties of multistep Runge-Kutta methods

This paper continues earlier work by the same author concerning the stability and B-convergence properties of multistep Runge-Kutta methods for the numerical solution of nonlinear stiff initial-value problems in a Hilbert space. A series of sufficient conditions and necessary conditions for a multistep Runge-Kutta method to be algebraically stable, diagonally stable, B- or optimally B-convergent are established, by means of which six classes of high order algebraically stable and B-convergent multistep Runge-Kutta methods are constructed in a unified pattern. These methods include the class constructed by Burrage in 1987 as special case, and most of them can be regarded as extension of the Gauss, RadauIA, RadauIIA and LobattoIIIC Runge-Kutta methods. We find that the classes of multistep Runge-Kutta methods constructed in the present paper are superior in many respects to the corresponding existing one-step Runge-Kutta schemes.