{"product_id":"general-linear-methods-for-ordinary-differential-equations-9780470408551","title":"General Linear Methods for Ordinary Differential","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eLearn to develop numerical methods for ordinary differential equations General Linear Methods for Ordinary Differential Equations fills a gap in the existing literature by presenting a comprehensive and up-to-date collection of recent advances and developments in the field.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface.  \u003cp\u003e\u003cb\u003e1 Differential Equations and Systems.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 The initial value problem.\u003c\/p\u003e \u003cp\u003e1.2 Examples of differential equations and systems.\u003c\/p\u003e \u003cp\u003e1.3 Existence and uniqueness of solutions.\u003c\/p\u003e \u003cp\u003e1.4 Continuous dependence on initial values and the right hand side.\u003c\/p\u003e \u003cp\u003e1.5 Derivatives with respect to parameters and initial values.\u003c\/p\u003e \u003cp\u003e1.6 Stability theory.\u003c\/p\u003e \u003cp\u003e1.7 Stiff differential equations and systems.\u003c\/p\u003e \u003cp\u003e1.8 Examples of stiff differential equations and systems.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Introduction to General Linear Methods.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Representation of general linear methods.\u003c\/p\u003e \u003cp\u003e2.2 Preconsistency, consistency, stage-consistency, and zero-stability.\u003c\/p\u003e \u003cp\u003e2.3 Convergence.\u003c\/p\u003e \u003cp\u003e2.4 Order and stage order conditions.\u003c\/p\u003e \u003cp\u003e2.5 Local discretization error of methods of high stage order.\u003c\/p\u003e \u003cp\u003e2.6 Linear stability theory of general linear methods.\u003c\/p\u003e \u003cp\u003e2.7 The types of general linear methods.\u003c\/p\u003e \u003cp\u003e2.8 Illustrative examples of general linear methods.\u003c\/p\u003e \u003cp\u003e2.9 Algebraic stability of general linear methods.\u003c\/p\u003e \u003cp\u003e2.10 Underlying one-step method.\u003c\/p\u003e \u003cp\u003e2.11 Starting procedures.\u003c\/p\u003e \u003cp\u003e2.12 Codes based on general linear methods.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Diagonally Implicit Multistage Integration Methods.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Representation of DIMSIMs.\u003c\/p\u003e \u003cp\u003e3.2 Representation formulas for the coefficient matrix \u003cb\u003eB\u003c\/b\u003e.\u003c\/p\u003e \u003cp\u003e3.3 A transformation for the analysis of DIMSIMs.\u003c\/p\u003e \u003cp\u003e3.4 Construction of DIMSIMs of type 1.\u003c\/p\u003e \u003cp\u003e3.5 Construction of DIMSIMs of type 2.\u003c\/p\u003e \u003cp\u003e3.6 Construction of DIMSIMs of type 3.\u003c\/p\u003e \u003cp\u003e3.7 Construction of DIMSIMs of type 4.\u003c\/p\u003e \u003cp\u003e3.8 Fourier series approach to the construction of DIMSIMs of high order.\u003c\/p\u003e \u003cp\u003e3.9 Least-squares minimization.\u003c\/p\u003e \u003cp\u003e3.10 Examples of DIMSIMs of type 1 and type 2.\u003c\/p\u003e \u003cp\u003e3.11 Nordsieck representation of DIMSIMs.\u003c\/p\u003e \u003cp\u003e3.12 Representation formulas for coefficient matrices \u003cb\u003eP\u003c\/b\u003e and \u003cb\u003eG\u003c\/b\u003e.\u003c\/p\u003e \u003cp\u003e3.13 Examples of DIMSIMs in Nordsieck form.\u003c\/p\u003e \u003cp\u003e3.14 Regularity properties of DIMSIMs.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Implementation of DIMSIMs.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Variable step size formulation of DIMSIMs.\u003c\/p\u003e \u003cp\u003e4.2 Local error estimation.\u003c\/p\u003e \u003cp\u003e4.3 Local error estimation for large step sizes.\u003c\/p\u003e \u003cp\u003e4.4 Construction of continuous interpolants.\u003c\/p\u003e \u003cp\u003e4.5 Step size and order changing strategy.\u003c\/p\u003e \u003cp\u003e4.6 Updating the vector of external approximations.\u003c\/p\u003e \u003cp\u003e4.7 Step-control stability of DIMSIMs.\u003c\/p\u003e \u003cp\u003e4.8 Simplified Newton iterations for implicit methods.\u003c\/p\u003e \u003cp\u003e4.9 Numerical experiments with type 1 DIMSIMs.\u003c\/p\u003e \u003cp\u003e4.10 Numerical experiments with type 2 DIMSIMs.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Two-Step Runge-Kutta Methods.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Representation of two-step Runge-Kutta methods.\u003c\/p\u003e \u003cp\u003e5.2 Order conditions for TSRK methods.\u003c\/p\u003e \u003cp\u003e5.3 Derivation of order conditions up to order six.\u003c\/p\u003e \u003cp\u003e5.4 Analysis of TSRK methods with one stage.\u003c\/p\u003e \u003cp\u003e5.5 Analysis of TSRK methods with two stages.\u003c\/p\u003e \u003cp\u003e5.6 Analysis of TSRK methods with three stages.\u003c\/p\u003e \u003cp\u003e5.7 Two-step collocation methods.\u003c\/p\u003e \u003cp\u003e5.8 Linear stability analysis of two-step collocation methods.\u003c\/p\u003e \u003cp\u003e5.9 Two-step collocation methods with one stage.\u003c\/p\u003e \u003cp\u003e5.10 Two-step collocation methods with two stages.\u003c\/p\u003e \u003cp\u003e5.11 TSRK methods with quadratic stability functions.\u003c\/p\u003e \u003cp\u003e5.12 Construction of TSRK methods with inherent quadratic stability.\u003c\/p\u003e \u003cp\u003e5.13 Examples of highly stable quadratic polynomials and TSRK methods.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Implementation of TSRK Methods.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Variable step size formulation of TSRK methods.\u003c\/p\u003e \u003cp\u003e6.2 Starting procedures for TSRK methods.\u003c\/p\u003e \u003cp\u003e6.3 Error propagation, order conditions, and error constant.\u003c\/p\u003e \u003cp\u003e6.4 Computation of approximations to the Nordsieck vector and local error estimation.\u003c\/p\u003e \u003cp\u003e6.5 Computation of approximations to the solution and stage values between grid points.\u003c\/p\u003e \u003cp\u003e6.6 Construction of TSRK methods with a given error constant and assessment of local error estimation.\u003c\/p\u003e \u003cp\u003e6.7 Continuous extensions of TSRK methods.\u003c\/p\u003e \u003cp\u003e6.8 Numerical experiments.\u003c\/p\u003e \u003cp\u003e6.9 Local error estimation of two-step collocation methods.\u003c\/p\u003e \u003cp\u003e6.10 Recent work on two-step collocation methods.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 General Linear Methods with Inherent Runge-Kutta Stability.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Representation of methods and order conditions.\u003c\/p\u003e \u003cp\u003e7.2 Inherent Runge-Kutta stability.\u003c\/p\u003e \u003cp\u003e7.3 Doubly companion matrices.\u003c\/p\u003e \u003cp\u003e7.4 Transformations between method arrays.\u003c\/p\u003e \u003cp\u003e7.5 Transformations between stability functions.\u003c\/p\u003e \u003cp\u003e7.6 Lower triangular matrices and characterization of matrices with zero spectral radius.\u003c\/p\u003e \u003cp\u003e7.7 Canonical forms of methods.\u003c\/p\u003e \u003cp\u003e7.8 Construction of explicit methods with IRKS and good balance between accuracy and stability.\u003c\/p\u003e \u003cp\u003e7.9 Examples of explicit methods with IRKS.\u003c\/p\u003e \u003cp\u003e7.10 Construction of A-stable and L-stable methods with IRKS.\u003c\/p\u003e \u003cp\u003e7.11 Examples of A-stable and L-stable methods with IRKS.\u003c\/p\u003e \u003cp\u003e7.12 Stiffly accurate methods with IRKS.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Implementation of GLMs with IRKS.\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Variable step size formulation of GLMs.\u003c\/p\u003e \u003cp\u003e8.2 Starting procedures.\u003c\/p\u003e \u003cp\u003e8.3 Error propagation for GLMs.\u003c\/p\u003e \u003cp\u003e8.4 Estimation of local discretization error and estimation of higher order terms.\u003c\/p\u003e \u003cp\u003e8.5 Computing the input vector of external approximations for the next step.\u003c\/p\u003e \u003cp\u003e8.6 Zero-stability analysis.\u003c\/p\u003e \u003cp\u003e8.7 Testing reliability of error estimation and estimation of higher order terms.\u003c\/p\u003e \u003cp\u003e8.8 Unconditional stability on nonuniform meshes.\u003c\/p\u003e \u003cp\u003e8.9 Numerical experiments.\u003c\/p\u003e \u003cp\u003e8.10 Local error estimation for stiffly accurate methods.\u003c\/p\u003e \u003cp\u003e8.11 Some remarks on recent work on GLMs.\u003c\/p\u003e \u003cp\u003eReferences.\u003c\/p\u003e \u003cp\u003eIndex.\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49525373141335,"sku":"9780470408551","price":123.45,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780470408551.jpg?v=1731860270","url":"https:\/\/bookcurl.com\/products\/general-linear-methods-for-ordinary-differential-equations-9780470408551","provider":"Book 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