Convergence of rational multistep methods of Adams-Padé type

• Published in: BIT (Nordisk Tidskrift for Informationsbehandling) Vol. 52; no. 1; pp. 3 - 20
• Main Authors: Auzinger, Winfried, Łapińska, Magdalena
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-03-2012; Springer
• Subjects: Approximations and expansions; Computational Mathematics and Numerical Analysis; Exact sciences and technology; Mathematical analysis; Mathematics; Mathematics and Statistics; Numeric Computing; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; 65L06; Evolution equations; Stiff initial value problems; 65L20; Rational multistep schemes; Adams schemes; Padé approximation; 65M12; Convergence; Matrix approximation; Rational approximation; Adams method; Differential equation; Stiff problem; Initial value problem; Evolution equation; Runge Kutta method; Partial differential equation; Extrapolation; Numerical analysis; Multistep method; Padé approximant
• ISSN: 0006-3835; 1572-9125
• DOI: 10.1007/s10543-011-0353-1

Rational generalizations of multistep schemes, where the linear stiff part of a given problem is treated by an
A
-stable rational approximation, have been proposed by several authors, but a reasonable convergence analysis for stiff problems has not been provided so far. In this paper we directly relate this approach to exponential multistep methods, a subclass of the increasingly popular class of exponential integrators. This natural, but new interpretation of rational multistep methods enables us to prove a convergence result of the same quality as for the exponential version. In particular, we consider schemes of rational Adams type based on
A
-acceptable Padé approximations to the matrix exponential. A numerical example is also provided.