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W. Auzinger, A. Eder, R. Frank:
"Extending convergence theory for nonlinear stiff problems, part II";
Report for ANUM Preprint No.22/01; 2001.

This paper deals with the convergence properties of high-order implicit Runge-Kutta methods applied to nonlinear stiff initial value problems. Earlier convergence concepts like the theory of B-convergence or singular perturbation analysis are not successfully applicable to stiff problems with a general 'geometry'. To overcome this drawback to a certain extent, in Part I of this paper a problem class was introduced, where the stiffness is axiomatically characterized in natural geometric terms, with a nonlinearly varying 'stiff eigendirection' corresponding to a 'stiff eigenvalue'. Furthermore, a convergence analysis for the implicit Euler scheme was presented.

In the meantime, some work has been done to extend this theory to general Runge-Kutta schemes. In particular, for the class of stiff problems considered in Part I, a complete analysis of Radau and Gauss schemes has been developed. In this paper these results are presented, commented, and the essential facts are proved. Due to their excellent stability properties, the Radau IIa schemes are of particular interest. Therefore we will work out the theory for these methods in some details, and content ourselves with reporting the corresponding results for Radau Ia and Gauss schemes, which are obtained in an analogous way.

Lack of space prevents us from presenting all the technical details, which can be found in an underlying report. In the proofs given, straightforward but lengthy algebraic manipulations are not always worked out in full detail. Our error estimates show that for certain stepsize ranges, the classical order of superconvergence can even be obtained in the stiff case. This is a further improvement compared to the B-convergence theory, where only the stage order is maintained in the error bounds.

http://www.math.tuwien.ac.at/~winfried/papers/anumpp2201.pdf