An Error Analysis Of Runge-kutta Convolution Quadrature
feedback return to old SpringerLink BIT Numerical MathematicsSeptember 2011, Volume 51, Issue 3, pp 483–496An error analysis of Runge–Kutta convolution quadratureAuthorsAuthors and affiliationsLehel BanjaiEmail authorChristian LubichArticleFirst Online: 25 January 2011Received: 26 May 2010Accepted: 30 December 2010DOI: 10.1007/s10543-011-0311-yCite this article as: Banjai, L. & Lubich, C. Bit Numer Math (2011) 51: 483. doi:10.1007/s10543-011-0311-y 14 Citations 265 Views AbstractAn error analysis is given for convolution quadratures based on strongly A-stable Runge–Kutta methods, for the non-sectorial case of a convolution kernel with a Laplace transform that is polynomially bounded in a half-plane. The order of approximation depends on the classical order and stage order of the Runge–Kutta method and on the growth exponent of the Laplace transform. Numerical experiments with convolution quadratures based on the Radau IIA methods are given on an example of a time-domain boundary integral operator.KeywordsConvolution quadratureRunge–Kutta methodsTime-domain boundary integral operatorsCommunicated by Timo Eirola.Mathematics Subject Classification (2000)65R2065L0665M15References1. Bamberger, A., Ha-Duong, T.: Formulation variationelle espace-temps pour le calcul par potentiel retardé d’une onde acoustique. Math. Methods Appl. Sci. 8, 405–435 (1986) MathSciNetMATHCrossRef2. Banjai, L.: Multistep and multistage convolution quadrature for the wave equation: algorithms and experiments. SIAM J. Sci. Comput. 32(5), 2964–2994 (2010) MathSciNetMATHCrossRef3. Banjai, L., Sauter, S.: Rapid solution of the wave equation in unbounded domains. SIAM J. Numer. Anal. 47(1), 227–249 (2008/2009) MathSciNetCrossRef4. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods. Wiley, New York (1987) MATH5. Calvo, M.P., Cuesta, E., Palencia, C.: Runge-Kutta convolution quadrature methods for well-posed equations with memory. Numer. Math. 107(4), 589–614 (2007) MathSciNetMATHCrossRef6. Dahlquist, G.: A special stability problem for linear multistep methods. BIT Numer. Math. 3, 27–43 (1963) MathSciNetMATHCrossRef7. Eggermont, P.P.B.: On the quadrature error in operational quadrature methods for convolutions. Numer. Math. 62(1), 35–48 (1992) MathSciNetMATHCrossRef8. Hackbusch, W., Kress, W., Sauter, S.A.: Sparse convolution quadrature for time domain boundary integral formulations of the wave equation. IMA J. Numer. Anal. 29(1), 158–179 (2009) MathSciNetMATHCrossRef9. Hairer, E., Wanner, G
convolution quadrature based on Runge-Kutta methodsArticle in Numerische Mathematik 133(4) · August 2015 with 9 ReadsDOI: 10.1007/s00211-015-0761-2 1st Maria Lopez-Fernandez18.34 · Sapienza University of Rome2nd S. SauterAbstractIn this paper, we develop the Runge-Kutta generalized convolution quadrature with variable time stepping for the numerical solution of convolution equations for time and space-time problems and present the corresponding stability and convergence analysis. For this purpose, some new theoretical tools such as tensorial divided differences, summation by parts with Runge-Kutta differences and a calculus for Runge-Kutta discretizations of generalized convolution operators such as an http://link.springer.com/content/pdf/10.1007/s10543-011-0311-y.pdf associativity property will be developed in this paper. Numerical examples will illustrate the stable and efficient behavior of the resulting discretization.Do you want to read the rest of this article?Request full-text CitationsCitations0ReferencesReferences16An error analysis of Runge–Kutta convolution quadrature[Show abstract] [Hide abstract] ABSTRACT: An error analysis is given for convolution quadratures based on strongly A-stable Runge–Kutta methods, for the non-sectorial https://www.researchgate.net/publication/281478177_Generalized_convolution_quadrature_based_on_Runge-Kutta_methods case of a convolution kernel with a Laplace transform that is polynomially bounded in a half-plane. The order of approximation depends on the classical order and stage order of the Runge–Kutta method and on the growth exponent of the Laplace transform. Numerical experiments with convolution quadratures based on the Radau IIA methods are given on an example of a time-domain boundary integral operator. KeywordsConvolution quadrature–Runge–Kutta methods–Time-domain boundary integral operators Full-text · Article · Sep 2011 Lehel BanjaiChristian LubichRead full-textGeneralized convolution quadrature with variable time stepping[Show abstract] [Hide abstract] ABSTRACT: In this paper, we will present a generalized convolution quadrature for solving linear parabolic and hyperbolic evolution equations. The original convolution quadrature method by Lubich works very nicely for equidistant time steps while the generalization of the method and its analysis to nonuniform time stepping is by no means obvious. We will introduce the generalized convolution quadrature allowing for variable time steps and develop a theory for its error analysis. This method opens the door for further development towards adaptive time stepping for evolution equations. As the m
Sie Ihren Suchbegriff ein: EnglishIDeutsch Hauptnavigation HomeThe InstituteResearchJobsIMPRSCalendarPeople Publications LibraryServicesInternal Navigation 2. Ebene Complete ListRecent Preprints http://www.mis.mpg.de/publications/preprints/2010/prepr2010-59.html Preprints Other seriesIndexNew preprintSupplemental MaterialPopular Science Navigation 3. Ebene 20162015201420132012201120102009200820072006200520042003200220012000199919981997 Preprint 59/2010Runge-Kutta convolution quadrature for operators arising in wave propagationLehel Banjai, Christian Lubich, and Jens Markus MelenkContact the author: Please use for correspondence this email.Submission date: 11. Oct. 2010Pages: 16published in: Numerische Mathematik, 119 (2011) an error 1, p. 1-20DOI number (of the published article): 10.1007/s00211-011-0378-z BibtexMSC-Numbers: 65R20, 65L06, 65M15Keywords and phrases: convolution quadrature, Runge-Kutta methods, Time-domain boundary integral operators, order reductionDownload full preprint: PDF (202 kB)Abstract:An error analysis of Runge-Kutta convolution quadrature is presented for a class of non-sectorial operators an error analysis whose Laplace transform satisfies, besides the standard assumptions of analyticity in a half-plane ℜs > σ0 and a polynomial bound O(sμ1) there, the stronger polynomial bound O(sμ2) in convex sectors of the form |arg s|≤ π∕2 - θ < π∕2 for θ > 0. The order of convergence of the Runge-Kutta convolution quadrature is determined by μ2 and the underlying Runge-Kutta method, but is independent of μ1. Time domain boundary integral operators for wave propagation problems have Laplace transforms that satisfy bounds of the above type. Numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour of Runge-Kutta convolution quadrature for this class of applications. Our results show in particular that the full classical order of the Runge-Kutta method is attained away from the scattering boundary. Preprints 20101/20102/20103/20104/20105/20106/20107/20108/20109/201010/201011/201012/201013/201014/201015/201016/201017/201018/201019/201020/201021/201022/201023/201024/201025/201026/201027/201028/201029/201030/201031/201032/201033/201034/201035/201036/201037/201038/201039/201040/201041/201042/201043/201044/201045/201046/201047/201048/201049/201050/201051/201052/201053/201055/201056/201057/201058/201059/