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Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations. (English) Zbl 0657.65093
This paper deals with the error analysis of implicit Runge-Kutta methods when they are applied to a special class of stiff differential systems. The systems under consideration, which arise in singular perturbations, have the form: \(y'=f(y,z)\), \((1)\quad \epsilon z'=g(y,z)\), where \(\epsilon\) \((>0)\) is a small parameter. The authors assume initial conditions such that (1) admits a unique solution with derivatives up to a sufficiently high order which are bounded independently of \(\epsilon\). Also f and g are sufficiently smooth and the logarithmic norm of \(g_ z\) is \(\leq -1.\)
By developping the exact and the approximate solutions in powers of \(\epsilon\) and using some results of Runge-Kutta methods applied to differential-algebraic systems, the authors have obtained asymptotic estimates for several classes of Runge-Kutta methods. These estimates have been checked numerically with the Van der Pol equation. Finally, it must be remarked that while the error analysis developed by R. Frank, J. Schneid and C. W. Ueberhuber [SIAM J. Numer. Anal. 22, 515-534 (1985; Zbl 0577.65056)] is concerned with more general stiff systems, the error bounds obtained here are generally sharper.
Reviewer: M.Calvo

MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65G50 Roundoff error
34A34 Nonlinear ordinary differential equations and systems
34E15 Singular perturbations for ordinary differential equations
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