Young Measure Approach to Computing Slowly Advancing Fast Oscillations

Artstein, Zvi; Linshiz, Jasmine S.; Titi, Edriss S.

doi:10.1137/070687219

Cited by 23 publications

(28 citation statements)

References 10 publications

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“…Notably, it is not always possible to obtain closed system giving the limit for the oscillatory dynamics or it may be even not clear how to split the dependent variables into the "slow" and "fast" parts. In these cases, an approach related with Young measures and the so-called slow observables may help to overcome the problem, see [3,4] for the details.…”

Section: Introductionmentioning

confidence: 99%

Large dispersion, averaging and attractors: three 1D paradigms

2018

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The effect of rapid oscillations, related to large dispersion terms, on the dynamics of dissipative evolution equations is studied for the model examples of the 1D complex Ginzburg-Landau and the Kuramoto-Sivashinsky equations. Three different scenarios of this effect are demonstrated. According to the first scenario, the dissipation mechanism is not affected and the diameter of the global attractor remains uniformly bounded with respect to the very large dispersion coefficient. However, the limit equation, as the dispersion parameter tends to infinity, becomes a gradient system. Therefore, adding the large dispersion term actually suppresses the non-trivial dynamics. According to the second scenario, neither the dissipation mechanism, nor the dynamics are essentially affected by the large dispersion and the limit dynamics remains complicated (chaotic). Finally, it is demonstrated in the third scenario that the dissipation mechanism is completely destroyed by the large dispersion, and that the diameter of the global attractor grows together with the growth of the dispersion parameter.2000 Mathematics Subject Classification. 35B40, 35B45.

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Section: Introductionmentioning

confidence: 99%

Large dispersion, averaging and attractors: three 1D paradigms

2018

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“…and it has a small but nonzero time average of −ǫΩ −1 C, which is however not reflected in the approximationX(t) = e ΩtȲ (t) = e Ωt X 0 obtained from classical averaging (9). It is easy to see, however, that if one uses a coordinate transform of Z = exp(−Ωt)(X + ǫΩ −1 C) instead of Y = exp(−Ωt)X, this nonzero average will be recovered, and the exact solution too.…”

Section: The Simple Improvementmentioning

confidence: 99%

Simply improved averaging for coupled oscillators and weakly nonlinear waves

Tao

2019

Communications in Nonlinear Science and Numerical Simulation

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The long time effect of nonlinear perturbation to oscillatory linear systems can be characterized by the averaging method, and we consider first-order averaging for its simplest applicability to high-dimensional problems. Instead of the classical approach, in which one uses the pullback of linear flow to isolate slow variables and then approximate the effective dynamics by averaging, we propose an alternative coordinate transform that better approximates the mean of oscillations. This leads to a simple improvement of the averaged system, which will be shown both theoretically and numerically to provide a more accurate approximation. Three examples are then provided: in the first, a new device for wireless energy transfer modeled by two coupled oscillators was analyzed, and the results provide design guidance and performance quantification for the device; the second is a classical coupled oscillator problem (Fermi-Pasta-Ulam), for which we numerically observed improved accuracy beyond the theoretically justified timescale; the third is a nonlinearly perturbed firstorder wave equation, which demonstrates the efficacy of improved averaging in an infinite dimensional setting.

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“…The method relies on the property that the value of the load determines the invariant measure. The latter assumption has been lifted in [ALT07], analyzing an example where the dynamics is not ergodic and the invariant measure for a given load is not unique, yet the invariant measure appearing in the limit dynamics can be detected by a good choice of an initial condition for the fast dynamics. The method is, again, to alternate between the computation of the invariant measure at a given time, say t, and using it then in the averaging needed to determine the slow equation.…”

Section: The Algorithmmentioning

confidence: 99%

Computing singularly perturbed differential equations

Chatterjee

Acharya

Artstein

2018

Journal of Computational Physics

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A computational tool for coarse-graining nonlinear systems of ordinary differential equations in time is discussed. Three illustrative model examples are worked out that demonstrate the range of capability of the method. This includes the averaging of Hamiltonian as well as dissipative microscopic dynamics whose 'slow' variables, defined in a precise sense, can often display mixed slow-fast response as in relaxation oscillations, and dependence on initial conditions of the fast variables. Also covered is the case where the quasi-static assumption in solid mechanics is violated. The computational tool is demonstrated to capture all of these behaviors in an accurate and robust manner, with significant savings in time. A practically useful strategy for accurately initializing short bursts of microscopic runs for the evolution of slow variables is integral to our scheme, without the requirement that the slow variables determine a unique invariant measure of the microscopic dynamics. * dl dt = L(x, l), which accommodates both a drift and a load. In the theoretical discussion we address the general case. We display the two particular cases, since there are many interesting examples of the type (2.1) or (2.2).An even more general setting would be the case where the right hand side of (2.3) is of the form H(x, l, ), namely, there is no a priori split of the right hand side of the equation into fast component and a drift or a slow component. A challenge then would be to identify, either analytically or numerically, such a split. We do not address this case here, but our study reveals what could be promising directions of such a general study.We recall that the parameter in the previous equation represents the ratio between the slow (or ordinary) part in the equation and the fast one. In Appendix B we examine one of our examples, and demonstrate how to derive the dimensionless equation with the small parameter, from the raw mechanical equation. In real world situations, is small yet it is not infinitesimal. Experience teaches us, however, that the limit behavior, as tends to 0, of the solutions is quite helpful in understanding of the physical phenomenon and in the computations. This is, indeed, demonstrated in the examples that follow.References that carry out a study of equations of the form (2.1) are, for instance, Tao, Owhadi and Marsden [TOM10], Artstein, Kevrekidis, Slemrod and Titi [AKST07], Ariel, Engquist and Tsai [AET09a, AET09b], Artstein, Gear, Kevrekidis, Slemrod and Titi [AGK + 11], Slemrod and Acharya [SA12]; conceptually similar questions implicitly arise in the work of Kevrekidis et al. [KGH + 03]. The form (2.2) coincides with the Tikhonov model, see, e.g., O'Malley [OJ14], Tikhonov, Vasileva and Sveshnikov [TVS85], Verhulst [Ver05], or Wasow [Was65]. The literature concerning this case followed, mainly, the so called Tikhonov approach, namely, the assumption that the solutions of the x-equation in (2.2), for l fixed, converge to a point x(l) that solves an algebraic equation, namely, the second equation ...

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Young Measure Approach to Computing Slowly Advancing Fast Oscillations

Cited by 23 publications

References 10 publications

Large dispersion, averaging and attractors: three 1D paradigms

Large dispersion, averaging and attractors: three 1D paradigms

Simply improved averaging for coupled oscillators and weakly nonlinear waves

Computing singularly perturbed differential equations

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