Analysis of the accuracy and convergence of equation-free projection to a slow manifold

Zagaris, Antonios; Gear, C. William; Kaper, Tasso J.; Kevrekidis, Yannis G.

ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 757-784.

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris, SIAM J. Appl. Dyn. Syst. 4 (2005) 711-732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the $m$th member of the class of algorithms ($m=0,1,...$) finds iteratively an approximation of the appropriate zero of the $(m+1)$st time derivative of the remaining variables and uses this root to approximate the location of the point on the slow manifold corresponding to these values of the observables. This article is the first of two articles in which the accuracy and convergence of the iterative algorithms are analyzed. Here, we work directly with fast-slow systems, in which there is an explicit small parameter, $\epsilon$, measuring the separation of time scales. We show that, for each $m=0,1,...$, the fixed point of the iterative algorithm approximates the slow manifold up to and including terms of $𝒪\left({\epsilon }^m\right)$. Moreover, for each $m$, we identify explicitly the conditions under which the $m$th iterative algorithm converges to this fixed point. Finally, we show that when the iteration is unstable (or converges slowly) it may be stabilized (or its convergence may be accelerated) by application of the Recursive Projection Method. Alternatively, the Newton-Krylov Generalized Minimal Residual Method may be used. In the subsequent article, we will consider the accuracy and convergence of the iterative algorithms for a broader class of systems - in which there need not be an explicit small parameter - to which the algorithms also apply.

[1] G. Browning and H.-O. Kreiss, Problems with different time scales for nonlinear partial differential equations. SIAM J. Appl. Math. 42 (1982) 704-718. | MR | Zbl

[2] J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences 35. Springer-Verlag, New York (1981). | MR | Zbl

[3] J. Curry, S.E. Haupt and M.E. Limber, Low-order modeling, initializations, and the slow manifold. Tellus 47A (1995) 145-161.

[4] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eq. 31 (1979) 53-98. | MR | Zbl

[5] C.W. Gear and I.G. Kevrekidis, Constraint-defined manifolds: a legacy-code approach to low-dimensional computation. J. Sci. Comp. 25 (2005) 17-28. | MR

[6] 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. | MR | Zbl

[7] S.S. Girimaji, Reduction of large dynamical systems by minimization of evolution rate. Phys. Rev. Lett. 82 (1999) 2282-2285.

[8] C.K.R.T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Montecatini Terme, L. Arnold Ed., Lecture Notes Math. 1609, Springer-Verlag, Berlin (1994) 44-118. | MR | Zbl

[9] H.G. Kaper and T.J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions. Physica D 165 (2002) 66-93. | MR | Zbl

[10] C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, Frontiers In Applied Mathematics 16. SIAM Publications, Philadelphia (1995). | MR | Zbl

[11] I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Commun. Math. Sci. 1 (2003) 715-762. | MR | Zbl

[12] H.-O. Kreiss, Problems with different time scales for ordinary differential equations. SIAM J. Numer. Anal. 16 (1979) 980-998. | MR | Zbl

[13] H.-O. Kreiss, Problems with Different Time Scales, in Multiple Time Scales, J.H. Brackbill and B.I. Cohen Eds., Academic Press (1985) 29-57. | MR | Zbl

[14] E.N. Lorenz, Attractor sets and quasi-geostrophic equilibrium. J. Atmos. Sci. 37 (1980) 1685-1699. | MR

[15] U. Maas and S.B. Pope, Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space. Combust. Flame 88 (1992) 239-264.

[16] P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics 107. Springer-Verlag, New York (1986). | MR | Zbl

[17] G.M. Shroff and H.B. Keller, Stabilization of unstable procedures: A recursive projection method. SIAM J. Numer. Anal. 30 (1993) 1099-1120. | MR | Zbl

[18] P. Van Leemput, W. Vanroose and D. Roose, Initialization of a Lattice Boltzmann Model with Constrained Runs. Report TW444, Catholic University of Leuven, Belgium (2005).

[19] P. Van Leemput, C. Vandekerckhove, W. Vanroose and D. Roose, Accuracy of hybrid Lattice Boltzmann/Finite Difference schemes for reaction-diffusion systems. Multiscale Model. Sim. 6 (2007) 838-857. | MR | Zbl

[20] A. Zagaris, H.G. Kaper and T.J. Kaper, Analysis of the Computational Singular Perturbation reduction method for chemical kinetics. J. Nonlin. Sci. 14 (2004) 59-91. | MR | Zbl

[21] A. Zagaris, H.G. Kaper and T.J. Kaper, Fast and slow dynamics for the Computational Singular Perturbation method. Multiscale Model. Sim. 2 (2004) 613-638. | MR | Zbl

[22] A. Zagaris, C. Vandekerckhove, C.W. Gear, T.J. Kaper and I.G. Kevrekidis, Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold. Numer. Math. (submitted).