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Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. (English)
Multiscale Model. Simul. 8, No. 4, 1269-1324 (2010).
This detailed paper introduces a new class of numerical methods based on averaging of the instantaneous flow of the following ordinary differential equation (ODE) on $\mathbb{R}^{d}$ $${\dot u}^ε = G(u^ε)+\tfrac{1}ε\,F(u^ε)$$ Such numerical integrators are called FLAVORS (FLow AVeraging integratORS) and enjoy some nice properties. They are multiscale; they do not need to identify explicitly or numerically the slow and fast variables; they can use a given numerical method as a black box to obtain solution at the microscopic time scale; they can be made to be symplectic, time-reversible, and symmetry preserving in all variables; they are explicit and applicable to stiff problems. About convergence, the notion of “two-scale flow convergence”, called $F$-convergence, is introduced and results about that are proved. Many numerical examples, including general Hamiltonian systems and the Fermi-Pasta-Ulam problem, are solved by the presented methods and well discussed.
Reviewer: Raffaella Pavani (Milano)