Optimal Balance via Adiabatic Invariance of Approximate Slow Manifolds

Abstract

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, to 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.

A comparison of methods to balance geophysical flows

“…Optimal balance is implemented by multiplying all nonlinear terms with a smooth monotonic 'ramp function' ρ(τ/T), where ρ : [0, 1] → [0, 1] with ρ(0) = 0 and ρ(1) = 1. Further, a sufficient number of derivatives of ρ need to vanish at the temporal endpoints; Gottwald et al (2017) give a rigorous analysis of why this is so. In this study, we use as ramp function…”

“…Further, a sufficient number of derivatives of need to vanish at the temporal endpoints; Gottwald et al. (2017) give a rigorous analysis of why this is so. In this study, we use as ramp function which was shown to yield asymptotically the best performance in Masur & Oliver (2020).…”

“…His argument presumes that the required integration is performed over an unbounded interval of time. Gottwald, Mohamad & Oliver (2017) studied optimal balance for the same finite-dimensional model restricted to a finite interval of time, which is necessary for any practical implementation. They realized that the required ramp function must satisfy consistency conditions at the temporal end points that preclude the use of analytic normal form theory for the mathematical analysis.…”

High-order uniformly accurate time integrators for semilinear wave equations of Klein-Gordon type in the non-relativistic limit

“…We refer the reader to [10,15] for the case of the Klein-Gordon equation and to Date: August 13, 2020. [9,12,13] for the case of system (2). Numerically, equation ( 1) is extensively studied in the relativistic regime [11,18].…”