The Method of Envelopes

The differential equation dx A , . where A = [x ~01 and e > 0 is a small parameter is a model for the stiff highly oscillatory problem. In this paper we discuss a new method for obtaining numerical approximations to the solution of the initial value problem for this differential equation. As e -» 0, the asymptotic theory for this initial value problem yields an approximation to the solution which develops on two time scales, a fast time t and a slow time r = t/e. We redevelop this asymptotic theory in such a form that the approximation consists of a series of simple functions of r, called carriers. (This series may be thought of as a Fourier series.) The coefficients of the terms of this series are functions of t. They are called envelopes and they modulate the carriers. Our computational method consists of determining numerical approximations to a finite collection of these envelopes. One of the principal merits of our method is its accuracy for the nonlinear problem. Introduction. The differential equation dx A , ,. v Tr-x+git,x), where A = [°x ~¿] and e > 0 is a small parameter is a model for the stiff highly oscillatory problem. In this paper we discuss a new method for obtaining numerical approximations to the solution of the initial value problem for this differential equation. As e —► 0, the asymptotic theory for this initial value problem yields an approximation to the solution which develops on two time scales, a fast time t and a slow time t = t/e (cf. Hoppensteadt-Miranker [11]). We redevelop this asymptotic theory in such a form that the approximation consists of a series of simple functions of t, called carriers. (This series may be thought of as a Fourier series.) The coefficients of the terms of this series are functions of t. They are called envelopes and they modulate the carriers. Our computational method consists of determining numerical approximations to a finite collection of these envelopes. This computational problem has been addressed by many others with a resulting variety of algorithms. A first class of algorithms consists of multistep methods which are exact for algebraic polynomials and/or trigonometric polynomials up to a certain degree (the degree may depend on the type of polynomial). We mention the work of Received February 11, 1977. AMS (MOS) subject classifications (1970). Primary 65L05, 34E99. *On leave from Free University of Amsterdam. ,.'i,pyright i§ 1978. American Mathematical Society 453 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 454 W. L. MIRANKER AND M. VAN VELDHUIZEN Gautschi [6], of Bertis and Stiefel, cf. [21] and the references therein, and the work of Snider-Fleming [20]. In the latter paper the modified multistep methods are also made more efficient by aliasing high frequencies by lower ones. Other methods have been introduced by Amdursky-Ziv [1], Hoppensteadt-Miranker [10] and MirankerWahba [15]. Amdursky and Ziv deflate the system, removing its highest frequency after its determination by computational means. Hoppensteadt and Miranker use the asymptotic theory, as they derive it in terms of infinite averages to develop numerical evaluation schemes. Miranker and Wahba compute running averages of the solution. (The envelope of the carrier, which itself is a constant, is, of course, the average.) Thus our method is related to and generalizes several of the existing attacks on this computational problem. One of the principal merits of our method is its accuracy for the nonlinear problem. In Section 1 we formulate the problem to be treated. We derive the asymptotic theory in the carrier-envelope form and we introduce and develop the notion of a smooth solution of a stiff differential equation. In Section 2 we show how the smooth solution of a differential equation may be approximated by a polynomial solution of an associated differential equation. We also introduce and give stability properties of backward differentiation formulae. These formulae are to be used later to determine the envelopes. Section 3 shows how complex valued carriers may be replaced by real ones and shows as well in what sense the smooth solution concept commutes with carrier changes. In Section 4 we describe two algorithms for generating envelopes. In Section 5 we demonstrate the existence and uniqueness of the solution produced by the computational scheme. We also obtain local and global error estimates as they depend on e, the mesh width h, the aliasing error and on other related parameters. A superconvergence result as well as stability with respect to perturbations in initial data are also obtained. All of these results in Section 5 are obtained without restriction on the ratio e/h. By restricting this ratio we are able to obtain an additional stability result, namely, stability with respect to roundoff errors. Finally in Section 6 we give results of computations with our methods on a sample nonlinear problem. We point out how the computations verify most of the theoretical behavior predicted in Section 5. 1. Exploration of the Problem. Consider the ordinary differential equation Here e > 0 is a small parameter, g is a smooth mapping from a domain in R x R2 into R2, x is an R2-valued map and A is the skew-symmetric matrix