A Multiscale Method for Highly Oscillatory Dynamical Systems Using a Poincaré Map Type Technique
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- Ariel, G., Engquist, B., Kim, S. et al. J Sci Comput (2013) 54: 247. doi:10.1007/s10915-012-9656-x
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We propose a new heterogeneous multiscale method (HMM) for computing the effective behavior of a class of highly oscillatory ordinary differential equations (ODEs). Without the need for identifying hidden slow variables, the proposed method is constructed based on the following ideas: a nonstandard splitting of the vector field (the right hand side of the ODEs); comparison of the solutions of the split equations; construction of effective paths in the state space whose projection onto the slow subspace has the correct dynamics; and a novel on-the-fly filtering technique for achieving a high order accuracy. Numerical examples are given.