AN EXTENSION OF B-CONVERGENCE FOR RUNGE-KUTTA METHODS

The well-known concepts of B-stability and B-convergence for the analysis of one-step methods applied to stiff initial value problems are based on the notion of one-sided Lipschitz continuity. In a recent paper (Auzinger et al. (1990)) the authors have pointed out that the one-sided Lipschitz constant m must often be expected to be very large (positive and of the order of magnitude of the stiff eigenvalues) despite a (globally) well-conditioned behavior of the underlying problem. As a consequence, the existing B-theory suffers from considerable restrictions; e.g., not even linear systems with time-dependent coefficients are satisfactorily covered. The purpose of the present paper is to fill this gap; for implicit Runge-Kutta methods we extend the B-convergence theory such as to be valid for a class of non-autonomous weakly nonlinear stiff systems; reference to the (potentially large) one-sided Lipschitz constant is avoided. Unique solvability of the system of algebraic equations is shown, and global error bounds are derived.