Abstract
This chapter introduces some interesting examples of differential equations and illustrates different types of qualitative behaviour of numerical methods. We deliberately consider only very simple numerical methods of orders 1 and 2 to emphasize the qualitative aspects of the experiments. The same effects (on a different scale) occur with more sophisticated higher-order integration schemes. The experiments presented here should serve as a motivation for the theoretical and practical investigations of later chapters. The reader is encouraged to repeat the experiments or to invent similar ones.
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References
This cat came to fame through Arnold (1963).
As is common in the study of mechanical problems, we use dots for denoting timederivatives, and we use primes for denoting derivatives with respect to other variables.
100 million years is not much in astronomical time scales; it just goes back to “Jurassic Park”.
We thank Alexander Ostermann, who provided us with this data.
Attention. In (3.4) and in the subsequent formulas qn denotes an approximation to q(nh), whereas qi in (3.1) denotes the ith subvector of q.
Irony of fate: Professor Loup Verlet, who later became interested in the history of science, discovered precisely “his” method in Newton’s Principia (Book I, figure for Theorem I). Private communication.
We are grateful to Prof. Ruth Durrer for helpful hints about this subject.
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© 2002 Springer-Verlag Berlin Heidelberg
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Hairer, E., Wanner, G., Lubich, C. (2002). Examples and Numerical Experiments. In: Geometric Numerical Integration. Springer Series in Computational Mathematics, vol 31. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05018-7_1
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DOI: https://doi.org/10.1007/978-3-662-05018-7_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-05020-0
Online ISBN: 978-3-662-05018-7
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