Stability and Dynamics of Numerical Methods for ...

IMA Journal of Numerical Analysis (1990) 10, 1-30

Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations A . ISERLES

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England [Received 19 January 1988]

1. Introduction DEVELOPMENTS in stability theory have contributed greatly in the last three decades to our understanding of behaviour of numerical methods for ordinary differential equations. The early theory—as is only to be expected—was linear, focusing on the scalar test equation y' = Xy, with _y(0) = l, where 9tA«O. In spite of the intrinsic simplicity of this model, it led to substantial insight into computational methods and to numerous constraints in their implementation. The two Dahlquist barriers (Henrici, 1962; Hairer et al, 1986) and the theory of order stars (Iserles & N0rsett, 1989, Wanner, 1987) manifest the power of linear analysis. However, stability theory came into its own with the formulation of the monotone (nonlinear) model by Dahlquist in the mid-seventies. Let the ODE system y'=f{t,y), y(to)=yo,

be given. Here / is a Lipschitz function and y e CN. We say that it is monotone if M(u-v,f(t,u)-f(t, t/)>«0 (r>/0), where (°, °) is an €2 inner product in C" and the inequality is valid for all u,v in a convex, non-empty, portion of C". Monotonicity implies dissipativity—given any two solutions u and v (with different initial conditions), the norm \\u(t) v(t)\\ is non-increasing in t—and it is of intense interest to verify whether this property is maintained by a numerical solution. Dahlquist (1978) showed that this is equivalent, in the case of one-leg methods, to the linear /1-stability and that this result can be easily translated to the more familiar multistep methods. A parallel

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Stability of numerical methods for nonlinear autonomous ordinary differential equations is approached from the point of view of dynamical systems. It is proved that multistep methods (with nonlinear algebraic equations exactly solved) with bounded trajectories always produce correct asymptotic behaviour, but this is not the case with Runge-Kutta. Examples are given of Runge-Kutta schemes converging to wrong solutions in a deceptively 'smooth' manner and a characterization of such two-stage methods is presented. PE(CE)m schemes are examined as well, and it is demonstrated that they, like Runge-Kutta, may lead to false asymptotics.

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A. ISERLES

be a given autonomous system of ordinary differential equations. We assume that / is a C1 function of y and set J(y) := df(y)/dy. Given that (1.1) has, for some initial value y0, finite asymptotics _p = lim^^ y(t), necessarily $ e F, where F :={y e CN :f(y) = 0} is the set of all complex zeros of/. We assume that (1.1) is being solved by a numerical method, which we consider as a general (M + 1-step) map

Here Yn approximates y(nh) and / i > 0 is the step-length. Obviously, if the numerical method has finite asymptotics Y = lim,^.,, Yn then ?eah:={YeCN:Y=

\imy(t) = $.

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A. ISERLES

We denote by P the linear stability domain of (2.1): V := {A e C : The solution of / = \y, y(0) = 1 by (2.1) tends to 0 as f-K»}. (2.4) Comparison of (2.3) and (2.4) yields at once the following. THEOREM

2 Y e Gh is asymptotically stable if o(/(/iY)) lim Yn = Y. (c) If an eigenvalue of J(Y) resides in the complement of the closure of V then no such neighbourhood V exists—but this does not exclude the existence of a lower-dimensional manifold of this kind. In particular, the choice of Yo = • • • = YM_X = Y leads to lim Yn = Y, regardless of attractivity of Y. (d) Absence of convergence is not the same as tendency to infinity! It is perfectly possible for the numerical solution sequence to cycle or to be chaotic when eigenvalues reside outside D. Useful insight into the asymptotic behaviour of (2.1) can be obtained from the scalar Riccati equation

where we assume that or,/3,yeC, y¥=0. Let $ = 1 ± [(1 — /3)2 — 4cry]i, where either sign will do, subject to $ # 0. Setting z{t) := Ay(t) + B, where A = - y/$ * 0, B = i(l — P/$), leads to the logistic equation z' = /8z(l-z), with the exact solution

Thus, F = {0, 1}.

z(0) = zo:=Ayo + B,

(2.5)

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(a) If Yn, for n = M, M + 1, . . . , tend to a bounded limit Y then Y is a correct asymptotic value of the underlying differential equation—although perhaps corresponding to a different initial value. (b) If o(/(/iY)) c T> then there exists a non-empty open neighbourhood K c C " x M such that

STABILITY AND DYNAMICS OF NUMERICAL METHODS

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Note that it is perfectly possible for y(t) to possess a polar singularity at finite t*, say. However, for t > t* the solution 'returns' from infinity and settles down, ultimately obeying the expected asymptotic behaviour. An identical linear transformation can be applied to any 'linear' numerical method: multistep, Runge-Kutta, etc. Thus, to derive numerical Riccati dynamics, it is enough to consider the logistic map (2.5). Henceforth, we restore the original notation y(t) and Yn for the exact solution and its numerical approximant respectively and use /3 instead of $. Figures 2.1 and 2.2 depict basins of attraction in /t/3 of the two fixed points 0 and 1 (i.e. the sets of /t/3 e C for which convergence to these values takes place), each for four different sets of starting values, Yo = • • • = YM e {0-5, 0-2, —0-2, —0-5}. Thus, each point corresponds to a choice of /3, the notation being ' + ' if lunY,,=0; ' x ' if limY,, = l; '»' if the sequence (Yn) is wandering—either period-doubling or chaotic. Outside these domains the sequence becomes unbounded. Note that, by Theorem 2, the theoretical basins of attraction of 0 and 1 are V and — D respectively. However, for most choices of starting values not all of ±D is attained and some choices of hfi in these sets lead to divergence. Figure 2.1 displays basins of attraction for forward Euler, i.e. one-step Adams-Bashforth. In that case V = {z e C : \z + 1| < 1}. Note that the portion of the /t/3 plane that corresponds to chaotic and period-doubling behaviour is negligible away from the real axis. This feature is consistent with the theory of iterated maps (Collet & Eckmann, 1980) and will appear in other figures in this paper. The onset of chaotic behaviour is the single feature of numerical ODE solvers that has attracted most attention of workers in dynamical systems (Prufer, 1985). Arguably, in the context of numerical analysis it is of minor interest only, since the complex plane, rather than just the real axis, need be considered in stability analysis of scalar equations to render results generalizable to systems of equations. Note that, as the maps z»->/3z(l-z) and z >->• z2 + /3(1 - /3) are topologically equivalent, the basins of attraction are linear maps of Mandelbrot sets (Blanchard, 1984). The multistep method in Fig. 2.2 is the three-step Adams-Bashforth. In that case V is crescent-shaped and B := D D -V ¥= 0 (c.f. Sand & 0sterby (1979) for a comprehensive survey of plots of linear stability domains). The ultimate 'destination' of /t/3 e B depends on starting values—it might be 0 or 1 or divergence. Again, wandering behaviour away from the real axis is not very pronounced, with the exception of the vicinity of the pure imaginary axis. This is, in fact, a welcome property: the exact solution for JR/3 = 0 is itself wandering and, to some extent, the numerical method mimics this behaviour. Instead of examining the /t/3 plane, it is perfectly possible to fix /3 and plot the basin of attraction in Y o eC, say (i.e. the portion of the Fatou set of the underlying map that corresponds to bounded fixed ponts). In the present framework, the /t/3 plane is more informative, because of its connection with the linear stability domain of the underlying method.

A. ISERLES

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FIG. 2.1. Forward Euler: Basin of attraction in hp, -3-2