B-series algebraic analysis of numerical methods / [electronic resource] :

• 作者： Butcher, John C.
• 其他作者：
  • SpringerLink (Online service)
• 其他題名：
  • Springer series in computational mathematics ;
• 出版： Cham : Springer International Publishing :Imprint: Springer
• 叢書名： Springer series in computational mathematics,v.55
• 主題： Runge-Kutta formulas. , Computational Mathematics and Numerical Analysis. , Numerical Analysis. , Analysis.
• ISBN： 9783030709563 (electronic bk.) 、 9783030709556 (paper)
• FIND@SFXID: CGU
• 資料類型: 電子書
• 內容註: 1 Differential equations, numerical methods and algebraic analysis -- 2 Trees and forests -- 3 B-series and algebraic analysis -- 4 Algebraic analysis and integration methods -- 5 B-series and Runge-Kutta methods -- 6 B-series and multivalue methods -- 7 B-series and geometric integration -- Answers to the exercises -- References -- Index.
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• 摘要註: B-series, also known as Butcher series, are an algebraic tool for analysing solutions to ordinary differential equations, including approximate solutions. Through the formulation and manipulation of these series, properties of numerical methods can be assessed. Runge-Kutta methods, in particular, depend on B-series for a clean and elegant approach to the derivation of high order and efficient methods. However, the utility of B-series goes much further and opens a path to the design and construction of highly accurate and efficient multivalue methods. This book offers a self-contained introduction to B-series by a pioneer of the subject. After a preliminary chapter providing background on differential equations and numerical methods, a broad exposition of graphs and trees is presented. This is essential preparation for the third chapter, in which the main ideas of B-series are introduced and developed. In chapter four, algebraic aspects are further analysed in the context of integration methods, a generalization of Runge-Kutta methods to infinite index sets. Chapter five, on explicit and implicit Runge-Kutta methods, contrasts the B-series and classical approaches. Chapter six, on multivalue methods, gives a traditional review of linear multistep methods and expands this to general linear methods, for which the B-series approach is both natural and essential. The final chapter introduces some aspects of geometric integration, from a B-series point of view. Placing B-series at the centre of its most important applications makes this book an invaluable resource for scientists, engineers and mathematicians who depend on computational modelling, not to mention computational scientists who carry out research on numerical methods in differential equations. In addition to exercises with solutions and study notes, a number of open-ended projects are suggested. This combination makes the book ideal as a textbook for specialised courses on numerical methods for differential
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