Computing singularly perturbed differential equations

Abstract

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 ...