Stability and B-convergence of linearly implicit Runge-Kutta methods
The investigation of B-convergence has been initiated by \textit{R. Frank}, \textit{J. Schneid} and \textit{C. W. Ueberhuber} [SIAM J. Numer. Anal. 18, 753-780 (1981; Zbl 0467.65032)] for implicit Runge-Kutta methods. They considered general stiff differential equations satisfying a one-sided Lipschitz condition. Since similar results cannot be expected for linearly implicit methods (such as W-methods, Rosenbrock methods,...), the author restricts his analysis to semi-linear problems, where the stiffness is present only in the linear part of the differential equation. This article presents a stability analysis which includes stability with respect to internal perturbations. This allows to prove that (under suitable stability assumptions) linearly implicit methods are B-convergent of order at least 1, i.e. the global error is O(h), where the constant in the O(.)-symbol is independent of the stiffness. It is left as an open question whether B-convergence of an order \(p>1\) can be achieved. For linear differential equations with constant coefficients a complete characterization of the order of B-convergence is given.
scientific article; zbMATH DE number 3988633
scientific article