Initial-Value Problems for ODE

  • Simon Širca
  • Martin Horvat
Part of the Graduate Texts in Physics book series (GTP)


The ability to reliably solve initial-value problems for ordinary differential equations is essential in order to understand the evolution of dynamical systems. In this chapter we deal with methods of advancing the given initial state of a system to later times, explaining clearly the role of stiffness, local discretization and round-off errors, and introducing the crucial concepts of consistency, convergence, and stability of difference schemes by which the differential equations are approximated. Single-step explicit and implicit methods are treated at length; automatic step-size control, embedding, and dense output are presented. Explicit and implicit multi-step (Adams, predictor–corrector, and backward differentiation) methods are analyzed from the viewpoint of stability and long-term integration. Runge–Kutta–Nyström methods are introduced as a special means to solve conservative second-order equations. Geometric and Lie series integrators are discussed as most viable options for high-precision integration requiring the preservation of certain invariants. The Examples and Problems include the Lorentz system, synchronization of globally coupled oscillators, chaotic scattering in two dimensions, problems in chemical kinetics, and a variety of astrophysical problems involving planets, stars and galaxies.


Symplectic Integrator Stiff Problem Implicit Euler Method Local Discretization Error Backward Differentiation Formula Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Simon Širca
    • 1
  • Martin Horvat
    • 1
  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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