Multiscale methods for stiff ODE: closure and consistency

Richard Sharp1, Richard Tsai2, and Björn Engquist2

1 Department of Mathematical Sciences, Carnegie Mellon University, USA.2 Department of Mathematics, The University of Texas at Austin, USA.

We apply a Heterogeneous Multiscale Method style algorithm to highly oscillatory ODEs. The algorithm tracks the slowevolution of a system in a manner consistent with, but more efficient than methods that resolve the fast oscillations. Anexample is given where local time averages of the original system variables form a closed set of macroscopic variables.

1 Introduction

Many systems of ordinary differential equations (ODEs) exhibit slow behavior that is the result of rapid oscillations. Asimple example, Kapitsa’s inverted pendulum, was considered in [1]. Rapid oscillations at the pivot stabilize the pendulum’supward fixed point allowing it to sway back and forth around the “twelve o’clock” position.

The slow dynamics of such systems may be efficiently modeled by a force function of the slow variables, but often thisfunction is unavailable. Multiscale techniques such as the Heterogeneous Multiscale Methods (HMM) calculate the slowmotion by using localized microscopic calculations to estimate the effective force which would be produced by the unknownmodel [2, 3]. The macroscopic system must contain sufficient information so that it is always possible to correctly initializethe microscopic system, but a gain in efficiency is still possible from only solving the microscopic system over short timeintervals. The Schematic in Section 2 illustrates the HMM approach applied to a two scale problem.

A set of slow variables is required to produce an HMM scheme. There is a trade-off between the simplicity of the macro-scopic variables, and the simplicity of the HMM algorithm which uses a particular set of macroscopic variables. The macro-scopic variables in this work are chosen for their simplicity. They are the time averages of the individual microscopic variables.Time averages are sufficient to capture the slow motion of linear and certain nonlinear systems. As explained in the next sec-tion, a usable collection of macroscopic variables must be closed. The resulting HMM algorithm must also be consistentwith the unique microscopic solution (which exists, but is not to be calculated) to produce a viable method. We present anon-obvious example in which consistency and closure of the macroscopic system may be proved.

Numerous techniques for approximating stiff ODEs are available, and many are available in texts for dissipative [4] andoscillatory [5] problems. With HMM, as with the geometric integrators, one expects correct averages, but not to exactlyreproduce the microscopic solution. A recent approach within the HMM framework by Ariel et al. [6] produces a simplerintegration scheme after a more complex search for macroscopic variables than the approach presented here. Also, the couplingof large and small time steps is an essential feature of the so called “equation free method” for stiff dissipative ODEs [7]. Theadvantage of the approach given here is the ability to apply it to challenging oscillatory problems with large scale variations.

2 Closure, Consistency, and an Example

An HMM algorithm attempts to replace an unknown macroscopic model with judicious use of microscopic calculations. Itis essential that the macroscopic model exists and is closed under the macroscopic variables being tracked by the algorithm.Closure in this setting is a property of the macroscopic variables which ensures that enough information is available to correctlyinitialize a microscopic calculation and evaluate the effective force. Closure does not imply that a given set of macroscopicvariables captures all of the slow dynamics of a system, only that those slow variables under consideration are self contained.Applied to the example below, {U} ≡ {0} is a closed set of slow variables that reveals nothing of interest about the system.

Given a set of slow variables, a basic HMM scheme for stiff ODEs proceeds, as shown in the Schematic, by,

1. The initial macroscopic conditions are given.

2. Initialize a compatible microscopic state.

3. Produce the microscopic solution on a short interval.

4. Verify the consistency of the microscopic solution: if not,update microscopic initial conditions and return to step 3;if so then continue to step 5.

5. Update the macroscopic state using the average micro-scopic force.

(5)

Microscale

Macroscale

time

(1)

(2)

(3)

(4)

Schematic

PAMM · Proc. Appl. Math. Mech. 7, 1140901–1140902 (2007) / DOI 10.1002/pamm.200700697

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The consistency check in step 4 is critical. An HMM algorithm produces simultaneous descriptions of a system at eachscale and both need to be consistent with the true solution. These two solutions must be consistent in the sense that themacroscopic trajectory produced by the method at each step is the same, to within a controllable error, as the macroscopictrajectory that would be extracted from the full microscopic solution, if it were known. Figure 1 provides an example ofconsistent and inconsistent solutions to a problem closely related to Equation 1. The true solution is depicted by a solid line.Two consistent HMM solutions converge to the true trajectory, while the inconsistent solution (the verification in step 4 wasnot performed) diverges and produces a smooth, but incorrect path. The consistency check is not needed in cases for whichthe procedure is formulated in an automatically consistent manner.

To demonstrate the ideas of consistency and closure, we will consider an oscil-

0.1

1

10

100

0 2 4 6 8 10

Log 1

0(A

vera

ge x

)

t

<x>(t)midpoint

EulerEuler vanilla

Fig. 1 Convergence of first and sec-ond order consistent schemes (circles andsquares), and divergence of an inconsis-tent scheme (triangles).

latory system for which the macroscopic variables are time averages of the micro-scopic variables. Time averaging is achieved by convolving the microscopic trajec-tories with a kernel K ∈ K

a,b of the kind presented in [3], U(t) = 〈u(·; ut′)〉 (t) =

limη→∞1

2η

∫ t+η

t−ηK(t − s)u(s)ds, whereu (·; ut′) is the local microscopic solution,

with initial condition ut′ , for some t′ ∈ [t − η, t + η]. Suppose uε(t) = f(t/ε)+g(t),where f is one periodic with zero mean and 0 < ε � 1, then the kernels in K

a,b elim-inate the oscillations, U = 〈uε〉 ≈ g(t). The effective force is, dU

dt= d〈uε〉

dt= 〈duε

dt〉,

and averages of higher moments may be used in a similar manner if they are neededto produce a closed set of macroscopic variables.

The following system produces a spiral solution which shrinks and shifts to theleft at a rate much slower than the rate at which it revolves about its center,

dx

dt= y

dy

dt= ε−1

(−x +

1

3

(−x +

√4x2 + 3εy2

))− y (1)

It is possible to construct a complete and consistent HMM scheme from the set of slow variables, {〈x〉 , 〈y〉}.

The key to describing the macroscopic evolution of the spiral is the slow variable r = 2

3

(−x +

√4x2 + 3εy2

), an

invariant if the system is altered by removing the −y term from dy

dt. For Equation 1, one may show that r ≈ 〈r〉. The

macroscopic equations become, d〈x〉dt

= 〈r〉 = − 〈x〉2

, d〈y〉dt

= 0 in the limit where ε → 0, η → 0, and ε/η → 0, establishing aclosure. Consistency is enforced, without special knowledge of the problem, by writing an error relation for the microscopicinitial conditions ut′ and updating (Steps 3 and 4) by Newton iteration. The process is efficient so the HMM scheme remainsfaster than brute force calculation (more so as scale separation increases). This requirement ensures that the microscopicinitial data, which may exhibit an O (1) discrepancy in comparison to the true microscopic solution, produces a macroscopicvalue equal to the true macroscopic value to within a bound controlled by ε and η. The decay rate, exp [−t/2] implied by thederived macroscopic equations is captured by the numerical solution.

The example described in the previous section is a model problem intended to demonstrate the usefulness of time averagesas slow variables for HMM in nonlinear oscillatory problems, as well as demonstrate the closure and consistency concepts.This approach has been applied to larger, more general systems for which a closed solution is not available. One successfulapplication has been to a chain of 20 masses connected by spring like forces.

Another important task for multiscale methods is to simplify complex microscopic systems through dimension reduction.Systems such as the chain, which may bunch up or become elongated, can be described by gross spatial characteristics. Anext step is to consider spatial averages of several microscopic variables in order to simplify the macroscopic system, bothconceptually and computationally, leading to a greater ability to capitalize on both aspects.

References

[1] R. Sharp, Y. H. Tsai, and B. Engquist, Multiple time scale numerical methods for the inverted pendulum problem, in: MultiscaleMethods in Science and Engineering, edited by B. Engquist, P. Lötstedt, and O. Runborg, Lecture Notes in Computational Scienceand Engineering Vol. 44 (Springer Verlag, 2005), pp. 241–261.

[2] W. E, Communications in Mathematical Sciences 1(3), 423–436 (2003).[3] B. Engquist and Y. H. Tsai, Mathematics of Computations 74, 1707–1742 (2005).[4] E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II, second edition, Springer Series in Computational Mathematics,

Vol. 14 (Springer-Verlag, Berlin, 1996), Stiff and differential-algebraic problems.[5] E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, Vol. 31

(Springer-Verlag, Berlin, 2002), Structure-preserving algorithms for ordinary differential equations.[6] G. Ariel, B. Engquist, and R. Tsai, A multiscale method for stiff ordinary differential equations with resonance, Tech. Rep. 07-17,

UCLA Center for Applied Mathematics, 2007, submitted to Mathematics of Computation.[7] I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg, and C. Theodoropoulos, Communications in Mathematical

Sciences 1(4), 715–762 (2003).

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