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Multiscale computations for highly oscillatory problems. (English) Zbl 1159.65332

Engquist, Björn (ed.) et al., Multiscale modeling and simulation in science. Papers based on the presentations at the summer school, Bosön, Stockholm, Sweden, June 2007. Berlin: Springer (ISBN 978-3-540-88856-7/pbk; 978-3-540-88857-4/ebook). Lecture Notes in Computational Science and Engineering 66, 237-287 (2009).
Summary: We review a selection of essential techniques for constructing computational multiscale methods for highly oscillatory ODEs. Contrary to the typical approaches that attempt to enlarge the stability region for specialized problems, these lecture notes emphasize how multiscale properties of highly oscillatory systems can be characterized and approximated in a truly multiscale fashion similar to the settings of averaging and homogenization. Essential concepts such as resonance, fast-slow scale interactions, averaging, and techniques for transformations to non-stiff forms are discussed in an elementary manner so that the materials can be easily accessible to beginning graduate students in applied mathematics or computational sciences.
For the entire collection see [Zbl 1155.65003].

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C29 Averaging method for ordinary differential equations
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