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G.A. Gottwald, H. Mohamad, and M. Oliver,
Optimal balance via adiabatic invariance of approximate slow
manifolds,
Multiscale Model. Simul. 15 (2017), 1404-1422.
Abstract:
We analyze the method of optimal balance which was introduced by
Viúdez and Dritschel (J. Fluid Mech. 521, 2004, pp. 343-352) to
provide balanced initializations for two-dimensional and
three-dimensional geophysical flows, here in the simpler context of a
finite dimensional Hamiltonian two-scale system with strong gyroscopic
forces. It is well known that when the potential is analytic, such
systems have an approximate slow manifold that is defined up to terms
that are exponentially small with respect to the scale separation
parameter. The method of optimal balance relies on the observation
that the approximate slow manifold remains an adiabatic invariant
under slow deformations of the nonlinear interactions. The method is
formulated as a boundary value problem for a homotopic deformation of
the system from a linear regime where the slow-fast splitting is known
exactly, and the full nonlinear regime. We show that, providing the
ramp function which defines the homotopy is of Gevrey class 2 and
satisfies vanishing conditions to all orders at the temporal end
points, the solution of the optimal balance boundary value problem
yields a point on the approximate slow manifold that is exponentially
close to the approximation to the slow manifold via exponential
asymptotics, albeit with a smaller power of the small parameter in the
exponent. In general, the order of accuracy of optimal balance is
limited by the order of vanishing derivatives of the ramp function at
the temporal end points. We also give a numerical demonstration of
the efficacy of optimal balance, showing the dependence of accuracy on
the ramp time and the ramp function.
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