A-Contractivity of Linearly Implicit Multistep Methods

Abstract

Previous article Next article A-Contractivity of Linearly Implicit Multistep MethodsRichard J. Charron and Min HuRichard J. Charron and Min Huhttps://doi.org/10.1137/0732011PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThe authors prove that there exist variable stepsize, variable coefficient, linearly implicit multistep methods of fixed but arbitrary order, which are A-contractive.[1] Hermann Brunner, Stabilization of optimal difference operators, Z. Angew. Math. Phys., 18 (1967), 438–444 36:1111 CrossrefISIGoogle Scholar[2] J. C. Butcher, The numerical analysis of ordinary differential equations, A Wiley-Interscience Publication, John Wiley & Sons Ltd., Chichester, 1987xvi+512, Runge-Kutta and General Linear Methods 88d:65002 0616.65072 Google Scholar[3] E. Hairer and , G. Wanner, Solving ordinary differential equations. II, Springer Series in Computational Mathematics, Vol. 14, Springer-Verlag, Berlin, 1991xvi+601, Stiff and Differential-Algebraic Problems 92a:65016 0729.65051 CrossrefGoogle Scholar[4] George D. Byrne and , Alan C. Hindmarsh, Stiff ODE solvers: a review of current and coming attractions, J. Comput. Phys., 70 (1987), 1–62 10.1016/0021-9991(87)90001-5 88e:65082 0614.65078 CrossrefISIGoogle Scholar[5] W. H. Hundsdorfer, Nonlinear stability analysis for a simple Rosenbrock method, Tech. Report, 81/31, Inst. of Appl. Math. and Comp. Sci., University of Leiden, Netherlands, 1981 Google Scholar[6] J. D. Lambert and , S. T. Sigurdsson, Multistep methods with variable matrix coefficients, SIAM J. Numer. Anal., 9 (1972), 715–733 10.1137/0709060 47:6095 0246.65024 LinkISIGoogle Scholar[7] Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr., 4 (1951), 258–281 13,254a 0042.12301 CrossrefGoogle Scholar[8] Olavi Nevanlinna and , Werner Liniger, Contractive methods for stiff differential equations. I, BIT, 18 (1978), 457–474 80h:65053a 0408.65045 CrossrefGoogle Scholar[9] Olavi Nevanlinna and , Werner Liniger, Contractive methods for stiff differential equations. II, BIT, 19 (1979), 53–72 80h:65053b 0408.65046 CrossrefGoogle Scholar[10] Syvert P. Nørsett, C-polynomials for rational approximation to the exponential function, Numer. Math., 25 (1975/76), 39–56 53:13939 0299.65010 CrossrefISIGoogle Scholar[11] Gary K. Rockswold, Implementation of $\alpha$-type multistep methods for stiff differential equations, J. Comput. Appl. Math., 22 (1988), 63–69 10.1016/0377-0427(88)90288-9 89d:65068 0643.65038 CrossrefISIGoogle Scholar[12] J. M. Sanz-Serna, Linearly implicit variable coefficient methods of Lambert-Sigurdsson type, IMA J. Numer. Anal., 1 (1981), 39–45 82f:65082 0465.65039 CrossrefISIGoogle Scholar[13] G. Wanner, , E. Hairer and , S. P. Nørsett, Order stars and stability theorems, BIT, 18 (1978), 475–489 81b:65070 0444.65039 CrossrefGoogle ScholarKeywordsmultistep methodsA-contractivity Previous article Next article FiguresRelatedReferencesCited byDetails Real-time Simulation Cross Ref Volume 32, Issue 1| 1995SIAM Journal on Numerical Analysis History Submitted:04 November 1991Accepted:15 September 1993Published online:14 July 2006 InformationCopyright © 1995 Society for Industrial and Applied MathematicsKeywordsmultistep methodsA-contractivityMSC codes65L0565L20PDF Download Article & Publication DataArticle DOI:10.1137/0732011Article page range:pp. 285-295ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics