W. Auzinger, R. Frank, G. Kirlinger:

"An extension of B-convergence for Runge-Kutta methods";

Applied Numerical Mathematics,9(1992), 91 - 109.

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 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.

http://dx.doi.org/10.1016/0168-9274(92)90008-2

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