Hans J. Stetter

Generalized Multistep Predictor-Corrector Methods

The order <italic>p</italic> which is obtainable with a stable <italic>k</italic>-step method in the numerical solution of <italic>y′</italic> = <italic>f</italic>(<italic>x</italic>, <italic>y</italic>) is limited to <italic>p</italic> = <italic>k</italic> + 1 by the theorems of Dahlquist. In the present paper the customary schemes are modified by including the value of the derivative at one “nonstep point;” as usual, this value is gained from an explicit predictor. It is shown that the order of these generalized predictor-corrector methods is not subject to the above restrictions; stable <italic>k</italic>-step schemes with <italic>p</italic> = 2<italic>k</italic> + 2 have been constructed for <italic>k</italic> ≤ 4. Furthermore it is proved that methods of order <italic>p</italic> actually converge like <italic>h<supscrpt>p</supscrpt></italic> uniformly in a given interval of integration. Numerical examples give some first evidence of the power of the new methods.

Global Error Estimation in Adams PC-Codes

Efficient and asymptotically (h --, 0) correct global error estimation procedures for initial value problem ODE-codes may be based on a suitable version of defect correction. An implementation of such a procedure for a variable order, variable step Adams PECE-code (typically STEP of Shampme) is described; the global error vector is produced simultaneously with the solution vector at each output point which need not be a grid point. The results of extensive experimentation are presented and d~scussed; m these tests, the qualitative and quantitative behavior of the true error was correctly reflected by the numerical estnnates throughout.

Numerical approximation of Fourier-transforms

ConclusionA method has been presented for the numerical evaluation of the integrals occuring in Fourier transformation which is based upon the approximation of the transform as a function of its variable co. The numerical information necessary for the construction of the approximation is gathered by the formation of alternating trapezoidal and rectangular sums without the use of trigonometric functions The case of a polynomial approximation has been elaborated in detail and numerical results have been presented.It is clear that using the same principle other types of approximating functions may be employed. In cases where the Fourier transform decreases faster than any power an exponential approximation may be effective. Further analysis and experimentation will serve to improve this seemingly powerful method.

Linear Multistep Methods

The structure of general m-stage k-step methods in the sense of Def. 2.1.8 and 2.1.10 is so complex that we will deal in this chapter only with the special class of one-stage k-step methods whose forward-step procedures consist simply of a linear combination of values of η μ and f (η μ ) at k + 1 consecutive gridpoints t µ , μ= v −k(1)v. No re-sub-stitutions into f (which formed the essence of RK-methods) will be permitted. Even with these limitations, this situation is sufficiently interesting and by no means trivial. Many of the results obtained will serve as background material in the treatment of more general classes of multistep methods in Chapter 5.

Runge-Kutta Methods

One-step methods (see Def. 2.1.8) form a particularly simple class of f. s. m. for IVP 1. Among these, a certain class of methods has commonly been associated with the names of C. Runge and W. Kutta and is widely used. These “Runge-Kutta methods” (RK-methods) are 1-step m+1-stage methods in the sense of Def. 2.1.10. However, only the final stage η v−1 m+1 of the value ηv−1 at tv−1 enters into the computation of η v ; also, only this final stage is normally taken as an approximation to the true solution. Thus, by formally disregarding the intermediate stages RK-methods may also be considered as 1-step 1-stage methods. We will use both interpretations depending on what is more convenient.

General Discretization Methods

In this introductory chapter we consider general aspects of discretization methods. Much of the theory is applicable not only to standard discretization methods for ordinary differential equations (both initial and boundary value problems) but also to a great variety of other numerical methods as indicated in the Preface (see also the end of Section 1.1.1). It should be emphasized that this is also true for the material on asymptotic expansions and their applications, although we have not elaborated on this. The chapter is concluded by a few remarks on the practical aspects of “solving” ordinary differential equations by discretization methods.

Forward Step Methods

In initial-value problems for ordinary differential equations the independent variable is commonly interpreted as time and the vector of unknown functions as a state vector varying in time. Given the state at one time instant t0, the desired solution consists of the state vectors during some time interval [t0, t0 + T].

Other Discretization Methods for IVP 1

In Chapters 3–5, we considered forward step procedures for IVP 1 which use evaluations of f only. In recent years it has been found that the automatic computation of values of higher derivatives of the local solution of an IVP 1 is feasible for large classes of IVP 1. In Section 6.1, we will survey some of the principal approaches which yield f. s. m. using such higher derivatives.

Multistage Multistep Methods

Having studied the peculiarities of multistage and multistep methods separately in Chapters 3 and 4 by analyzing their simplest representative classes we will now consider discretization methods for IVP1 which combine the features of multistage and multisteps methods, cf. Section 2.1.3. We will, however, still restrict ourselves to forward step methods which compute only quantities which are supposed to be approximations to values of z and z′; other f. s. m. will be discussed in Chapter 6. The emphasis of the analysis will be on some important special classes of multistage multistep methods.

Wesen und Ziel der Informatik

Die Wissenschaft, fur die im deutschen Sprachgebiet seit kurzem die Bezeichnung „Informatik“ ublich geworden ist, wird im angelsachsischen Bereich meist als „Computer Science” bezeichnet, sie steht also offensichtlich in sehr engem Zusammenhang mit dem Computer. Nun ist die Existenz der Computer zwar fur jeden Osterreicher eine Tatsache, die ihm an den verschiedensten Beispielen fast taglich vor Augen gefuhrt wird, ob er nun Erlagscheine in Lochkartenform oder vom Computer erstellte Lohnzettel erhalt. Trotzdem umgibt den Computer in den Augen der meisten Mitburger ein mystischer Schein, sodas derzeit alle moglichen Vorgange in den Augen der Offentlichkeit allein dadurch gerechtfertigt werden konnen, das sie mit Hilfe eines Computers abgewickelt werden. (Das ist ebenso unsinnig wie wenn die Qualitat oder gar die Existenz eines industriellen Produkts durch seine Herstellung am Fliesband gerechtfertigt wurde.) Etwas konkreter stellt sich der Laie unter einem Computer eine riesige Rechenmaschine, einen gewaltigen Buchhaltungsautomaten oder ein kompliziertes elektronisches Schaltungsgebilde vor. Waren dies die einzigen Kennzeichen eines Computers, so musten die Mathematik, die Betriebswirtschaftslehre und die Nachrichtentechnik ausreichen, seinen Einsatz wissenschaftlich zu behandeln.