Abstract
This chapter deals with the numerical solution of initial value problems of ordinary differential equations. First, different approaches to obtain integration methods for ordinary differential equations are presented. Then the important properties consistency, stability, and convergence of integration methods are introduced and studied. This allows for a fruitful application of the results obtained in Chapter 12. Subsequently, one-step methods are considered in detail and their important properties are shown. Moreover, Runge–Kutta methods are thoroughly investigated. Finally, we deal with linear multistep methods and study the asymptotic behavior of integration methods and of stiff differential equations. In particular, we deal with Dahlquist’s root condition and derive the first Dahlquist barrier and the second Dahlquist barrier.
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© 2016 Springer International Publishing Switzerland
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Römisch, W., Zeugmann, T. (2016). Numerical Solution of Ordinary Differential Equations. In: Mathematical Analysis and the Mathematics of Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-42755-3_13
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DOI: https://doi.org/10.1007/978-3-319-42755-3_13
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42753-9
Online ISBN: 978-3-319-42755-3
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