Abstract
Existing convergence concepts for the analysis of discretizations of nonlinear stiff problems suffer from considerable drawbacks. Our intention is to extend the convergence theory to a relevant class of nonlinear problems, where stiffness is axiomatically characterized in natural geometric terms.
Our results will be presented in a series of papers. In the present paper (Part I) we motivate the need for such an extension of the existing theory, and our approach is illustrated by means of a convergence argument for the Implicit Euler scheme.
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W. Auzinger, R. Frank, and G. Kirlinger,Extending convergence theory for nonlinear stiff problems. Part I, Report 116/94, Institute for Applied and Numerical Mathematics, Vienna University of Technology (extended version of the present paper).
W. Auzinger, R. Frank, and G. Kirlinger,A note on convergence concepts for stiff problems, Computing 44 (1990), pp. 197–208.
K. Dekker and J. G. Verwer,Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, 1984.
R. Frank and C. W. Ueberhuber,Iterated defect-correction for the efficient solution of stiff systems of ordinary differential equations, BIT 17 (1977), pp. 146–159.
R. Frank, J. Schneid, and C.W. Ueberhuber,The concept of B-convergence, SIAM J. Numer. Anal. 18 (1981), pp. 753–780.
R. Frank, J. Schneid, and C. W. Ueberhuber,Stability properties of implicit Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985), pp. 497–515.
R. Frank, J. Schneid, and C. W. Ueberhuber,Order results for implicit Runge-Kutta methods applied to stiff systems, SIAM J. Numer. Anal. 22 (1985), pp. 515–534.
E. Hairer, C. Lubich, M. Roche,Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations, BIT 28 (1988), pp. 678–700.
E. Hairer and G. Wanner,Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, 1991.
Ch. Lubich,On the convergence of multistep methods for nonlinear stiff differential equations, Numer. Math. 58 (1991), pp. 839–853; Erratum, 61 (1992), pp. 277–279.
K. Nipp and D. Stoffer,Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type, Numer. Math., 70 (1995), pp. 245–257.
A. Prothero and A. Robinson,On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comp. 28 (1974), pp. 145–162.
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Auzinger, W., Frank, R. & Kirlinger, G. Extending convergence theory for nonlinear stiff problems part I. Bit Numer Math 36, 635–652 (1996). https://doi.org/10.1007/BF01733784
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DOI: https://doi.org/10.1007/BF01733784