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**Numerical solution of highly oscillatory ordinary differential equations.**
*(English)*
Zbl 0887.65072

Iserles, A. (ed.), Acta Numerica Vol. 6, 1997. Cambridge: Cambridge University Press. 437-483 (1997).

The authors consider systems whose solutions may be oscillatory in the sense that there is a fast solution which varies regularly about a slow solution. These problems are called highly oscillatory if the time scale of the fast solution is much shorter than the interval of integration. Sometimes one might be interested only in finding the slow solution, in other situations it may be important to recover more information about the high-frequency oscillation, such as its amplitude, its energy or its envelope.

Form and structure of the oscillating problem as well as the selection of an appropriate numerical method are highly application-dependent. A wide variety of numerical methods has been developed for highly oscillatory problems. In Section 2 of this interesting paper the authors discuss basic concepts on numerical methods for linear highly oscillatory problems and characterize a lot of special numerical methods developed since 1960.

Small-amplitude oscillations in linear or nearly linear systems can often be damped via highly stable implicit numerical methods. It is also feasible to damp the oscillation in certain structured, highly nonlinear oscillating problems from mechanical systems. Even with numerical methods based simply on damping the oscillations, there can be unforeseen difficulties due to the nonlinear oscillation for example in automatic stepsize control and in obtaining convergence of the Newton iteration of implicit numerical methods. It is important to recognize that, in general one should not expect to be able to numerically solve nonlinear highly oscillatory problems using stepsizes which are large relative to the timescale of the fast solution.

The authors discuss some classes of application problems. Special interest is focused in highly oscillatory rigid and flexible mechanical systems, describing the nonlinear structure of these systems and implications for numerical methods, when and how the oscillation can be safely and efficiently damped, modal analysis techniques from structural analysis, and the problems in extending these techniques to flexible multibody systems. Problems and numerical analysis for molecular dynamics, circuit analysis and orbital mechanics are also described.

For the entire collection see [Zbl 0868.00024].

Form and structure of the oscillating problem as well as the selection of an appropriate numerical method are highly application-dependent. A wide variety of numerical methods has been developed for highly oscillatory problems. In Section 2 of this interesting paper the authors discuss basic concepts on numerical methods for linear highly oscillatory problems and characterize a lot of special numerical methods developed since 1960.

Small-amplitude oscillations in linear or nearly linear systems can often be damped via highly stable implicit numerical methods. It is also feasible to damp the oscillation in certain structured, highly nonlinear oscillating problems from mechanical systems. Even with numerical methods based simply on damping the oscillations, there can be unforeseen difficulties due to the nonlinear oscillation for example in automatic stepsize control and in obtaining convergence of the Newton iteration of implicit numerical methods. It is important to recognize that, in general one should not expect to be able to numerically solve nonlinear highly oscillatory problems using stepsizes which are large relative to the timescale of the fast solution.

The authors discuss some classes of application problems. Special interest is focused in highly oscillatory rigid and flexible mechanical systems, describing the nonlinear structure of these systems and implications for numerical methods, when and how the oscillation can be safely and efficiently damped, modal analysis techniques from structural analysis, and the problems in extending these techniques to flexible multibody systems. Problems and numerical analysis for molecular dynamics, circuit analysis and orbital mechanics are also described.

For the entire collection see [Zbl 0868.00024].

Reviewer: H.Ade (Mainz)

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

70F10 | \(n\)-body problems |

70M20 | Orbital mechanics |

81V55 | Molecular physics |

34A34 | Nonlinear ordinary differential equations and systems |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |