ON SOME ONE-STEP IMPLICIT METHODS AS DYNAMICAL SYSTEMS

. The one-step implicit methods, the backward Euler being the most known, require the solution of a nonlinear equation at each step. To avoid this, these methods can be approximated by making use of a one step of a Newton method. Thus the methods are transformed into some explicit ones. We will obtain these transformed methods, ﬁnd conditions under which they generate continuous dynamical systems and show their order of convergence. Some results on the stability of these explicit schemes, as well as on the shadowing phenomenon are also carried out. Concluding remarks and some open problems end the paper.


INTRODUCTION
We consider the initial value autonomous problem (Cauchy problem): (1) where u (t) ∈ R p denotes a vector valued function of t ∈ R, and f ∈ C (R p , R p ).Further regularity assumptions on f will be described as required.The aim of this paper is to solve the above problem by a method which avoids the solution of a nonlinear equation at each step.We will work in the spirit of dynamical systems and will use the notations from the monograph of Stuart and Humphries [9].In fact, we use one step of a Newton method in order to transform a one-step implicit method into an explicit one.
We will consider the following two classes of implicit schemes: A/ the one-stage theta method and B/ the two-stage theta method Note that both schemes A/ and B/ reduce to the forward Euler method when θ = 0 and to the backward Euler method when θ = 1.However, for θ ∈ (0, 1) the methods differ.For example, if θ = 1 2 in A/ we obtain the implicit midpoint rule while if θ = 1  2 in B/ we get the trapezoidal rule.More important than that, whenever the schemes A/ and B/ are implicit they require at each step of time n the solution of a nonlinear system.To avoid this, the method is frequently approximated by applying one step of Newton iteration, with U n as an initial guess.Consequently, we transform both, A/ and B/ into explicit schemes and find out conditions under which they are continuous dynamical systems.This is the subject of the second section.In the third section we analyse them with respect to the order of convergence.In the fourth section we consider the shadowing phenomenon and the numerical stability.

THE NEWTON METHOD FOR THE ONE-STAGE AND TWO-STAGE THETA METHODS
It is quite surprising that when the classical Newton method is used in order to solve the implicit equations involved by both A/ and B/, with U n as initial guess, we are lead up to the same explicit scheme, namely where h := ∆t, and the increment function is defined by with df (•) denoting the Jacobian of f (•) .
It is a matter of evidence (see for example [2] and [7]) that for the matrix I p − hθdf (U n ) , has a continuous inverse and the Neumann series Thus under the condition (2) for hθ, AB/ defines a dynamical system on R p with the discrete semigroup of evolution Moreover, if f (•) is continuously differentiable, the above dynamical system is itself continuous.

THE ORDER OF CONVERGENCE OF THE TRANSFORMED EXPLICIT METHOD
The truncation error for the scheme AB/, as a numerical approximation of the problem (1), in a point U of R p is defined by ∆t U, where S (•) : R p → R p is the evolution semigroup associated with (1), (see for example [9, p. 106] and [10]).Taylor expansion of S (∆t) U shows that there exists ∆t c = ∆t c (U ) > 0 and K = K (U ) such that We have to observe that K is the upper bound for the principal error function We also note that, due to the terms d 2 f (•), f 2 in d 3 u, no choice of the parameter θ would imply a method of the third order of accuracy.
For θ = 1 2 in AB/ the truncation error satisfies an inferior condition, namely where is an upper bound for the principal error function Consequently, the implicit midpoint rule and the trapezoidal rule are of order of accuracy 2, while any other methods are of order 1.

ON THE SHADOWING PHENOMENON AND THE NUMERICAL STABILITY OF THE EXPLICIT SCHEME
The idea of approximating a solution of (1) by a numerical scheme with different initial data is sometimes referred to as shadowing.
Thus, for hθ satisfying (2), let use AB/ in order to solve (1) with two different initial data U and V, U = V.We have respectively Subtracting these two equations, we get Whenever f (•) satisfies a Lipschitz condition with the Lipschitz constant denoted by L f , the last equation implies successively Consequently, assuming that f (•) is Lipschitz and (1) generates a dynamical system on R p , the inequality (6) ensure the existence of the local phase portrait of (1) near a critical point (see [9,Ch. 6

: Convergence of Invariant Sets]).
A quite exhaustive analysis of the linear stability in the numerical solution of initial value problems is available in [3].When the dynamical system is nonlinear the analysis becomes much more complicated.
With respect to the stability of the scheme AB/ we have obtained the following estimation ( 7) where N is the sum of the Neumann series whenever hθ satisfies (2).The estimation (7) could be quite useful for vector fields f (•) with some boundedness properties.

CONCLUDING REMARKS AND OPEN PROBLEMS
As it is apparent from the above analysis the most important results obtained-the existence of the dynamical system, the stability and the shadowing results-depend essentially on the inequality (2).Thus, whenever the Jacobian df (U n ) , n = 1, 2, 3, . .., is a normal matrix, its eigenvalues can be used in order to verify (2).Unfortunately, in the most interesting, and at the same time most important applications, df (•) is not a normal operator (matrix).In this situation, the spectrum of this operator could be misleading (see for example [1], [4], [11] or [12]).The pseudospectrum and some scalar measure of non-normality of square matrices, such the Henrici number [5], could have some relevance.However, the two parameters h and θ involved in the inequality (2) can be adjusted, in principle, such that this inequality is satisfied at each and every step.
A final remark about the terminology used in this paper.Finite difference schemes for initial value problems (1) are considered in the classical language in the well known monographs of Henrici [6], Richtmyer and Morton [8] or in the textbook of Süli and Meyers [10].In contrast with this language is that of dynamical systems based essentially on semigroups of evolution.We have chosen the latter due to a multitude of advantages.
At the end of this paper we have to mention an important open problem.This refers to the specialization of the scheme AB/, and the subsequent results, for some specific vector fields f (•) such as gradient, dissipative and contractive vector fields.