Seminar in Numerical Analysis: Daniel Weiss (Universität Tübingen)

A Heterogeneous Multiscale Method for highly-oscillatory Hamiltonian systems with solution-dependent frequencies

There are many systems which exhibit different scales which act in the following way: A slow dynamic of interest is driven or affected by highly-oscillatory components of the systems. A simple example is an old-fashioned alarm clock, which moves on a table driven by the fast movements of the clapper.

We will explain the Heterogeneous Multiscale Method (HMM)[1], which is believed to provide a numerical method for all kind of multiscale systems to overcome the difficulties of numerical integration generated by highly-oscillatory components. We will formulate the HMM for highly-oscillatory Hamiltonian systems with solution-dependent frequencies more precisely for the double spring pendulum with very stiff springs. Finally we will discuss the drawbacks of this method in case of solution-dependent frequencies.

[1] E, W.; Engquist, B.: The heterogeneous multi-scale method, Comm. Math. Sci., 1, 87--133, 2003.

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