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General linear methods for ordinary differential equations. (English) Zbl 1211.65095

Hoboken, NJ: John Wiley & Sons (ISBN 978-0-470-40855-1/hbk; 978-0-470-52216-5/ebook). xv, 482 p. (2009).
In Chapter 1, the author presents a short introduction to ordinary differential equations (ODEs), including existence and uniqueness theory, continuous dependence on the initial data and right hand side, stability theory and discussion of stiff differential equations and systems.
In Chapter 2, he gives an introduction to general linear methods. In this chapter, he discusses pre-consistency, consistency, stage-consistency, zero stability, convergence, order and stage order conditions and linear stability theory for non-stiff and stiff differential systems.
In Chapter 3, different types of diagonally implicit multi-stage integration methods (DIMSIMs) are discussed. The various results related to DIMSIM by Butcher and Jackiewiez are discussed. The construction of all the four types of DIMSIMs with the same desirable stability property and also with appropriate stage order and order conditions with examples are discussed. Fourier series method for the construction of higher order DIMSIMs are also discussed.
In Chapter 4, various practical issues related to the implementation of DIMSIMs such as variable step size formulation, the choice of the initial step size and the initial order of integration, the error propagation and estimation of the principal part of the local discretization error for small and large step lengths, construction of continuous interpolation, step size and order changing strategies using the Nordsieck vector representation are discussed. Further the local discretization error for sufficiently smooth function is discussed with examples of explicit and implicit formulas with small and large step sizes. Numerical experiments with type 1 and type 2 DIMSIMs are also discussed.
In Chapter 5, the author discusses the general class of different types of explicit and implicit Runge-Kutta (TSRK) methods with local discretization errors, with different stage orders. The author also discusses the solution strategies for large system of ODEs and integro-differential equations. Further, a two-step collocation method for the numerical solution of initial value problems, stability and quadratic stability with examples are also discussed in this chapter.
In Chapter 6, the formulation of TSRK of various order, computations of approximations to the Nordsieck vector and its local error estimation, continuous extension of TSRK method and numerical experiments of various TSRK methods are discussed in this chapter.
In Chapter 7, general linear methods (GLMs) with inherent Runge-Kutta stability (IRKS) discussed by Butcher and Wright are included. The author also discusses doubly companion matrices, transformation between method arrays, transformations between stability functions and canonical forms of methods in this chapter. Constructions of explicit GLMs with IRKS and a good balance between accuracy and stability with examples are also demonstrated. Construction and examples of A- and L-stable GLMs with IRKS are also discussed in this chapter.
In Chapter 8, implementation of GLMs with IRKS, and error propagation of GLMs of certain order and stage order are incorporated. Estimation of the local discretization error and testing its reliability, zero stability analysis, and unconditional stability on non-uniform meshes are discussed in this chapter. Local error estimation for stiffly accurate GLMs with IRKS are also discussed in this chapter.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
34A34 Nonlinear ordinary differential equations and systems
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
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