A multiscale method for highly oscillatory dynamical systems using a Poincaré map type technique. (English) Zbl 1263.65067
Summary: We propose a new heterogeneous multiscale method 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.
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
Keywords:
oscillatory dynamical system; averaging; multiscale method; nonstandard splitting; filtering; numerical examplesReferences:
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