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Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold

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  • In [C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes, SIAM J. Appl. Dyn. Syst. 4 (2005), 711--732], we developed the family of constrained runs algorithms to find points on low-dimensional, attracting, slow manifolds in systems of nonlinear differential equations with multiple time scales. For user-specified values of a subset of the system variables parametrizing the slow manifold (which we term observables and denote collectively by $u$), these iterative algorithms return values of the remaining system variables $v$ so that the point $(u,v)$ approximates a point on a slow manifold. In particular, the $m-$th constrained runs algorithm ($m = 0, 1, \ldots$) approximates a point $(u,v_m)$ that is the appropriate zero of the $(m+1)-$st time derivative of $v$. % The accuracy with which $(u,v_m)$ approximates the corresponding point on the slow manifold with the same value of the observables has been established in [A. Zagaris, C. W. Gear, T. J. Kaper and I. G. Kevrekidis, Analysis of the accuracy and convergence of equation-free projection to a slow manifold, ESAIM: M2AN 43(4) (2009) 757--784] for systems for which the observables $u$ evolve exclusively on the slow time scale. There, we also determined explicit conditions under which the $m-$th constrained runs scheme converges to the fixed point $(u,v_m)$ and identified conditions under which it fails to converge. Here, we consider the questions of stability and stabilization of these iterative algorithms for the case in which the observables $u$ are also allowed to evolve on a fast time scale. The stability question in this case is more complicated, since it involves a generalized eigenvalue problem for a pair of matrices encoding geometric and dynamical characteristics of the system of differential equations. We determine the conditions under which these schemes converge or diverge in a series of cases in which this problem is explicitly solvable. We illustrate our main stability and stabilization results for the constrained runs schemes on certain planar systems with multiple time scales, and also on a more-realistic sixth order system with multiple time scales that models a network of coupled enzymatic reactions. Finally, we consider the issue of stabilization of the $m-$th constrained runs algorithm when the functional iteration scheme is divergent or converges slowly. In that case, we demonstrate on concrete examples how Newton's method and Broyden's method may replace functional iteration to yield stable iterative schemes.
    Mathematics Subject Classification: Primary: 65L11, 65L20; Secondary: 15A22, 34E05, 37M99, 92C45.


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