H. A. Watts

Solving Nonstiff Ordinary Differential Equations—The State of the Art

The characteristics and capabilities of the best codes for solving the initial value problem for ordinary differential equations are studied. Only codes which are readily available, portable, and v...

Global Error Estimates for Ordinary Differential Equations

The user of a code for solving the initial value problem for ordinary differential equations is normally interested in the global error, i.e. the difference between the solution of the problem posed and the numerical result returned by the code. This paper describes a way of estimating the global error reliably while still solving the problem with acceptable efficiency. Global extrapolation procedures are applied to parallel solutions obtained by a Runge-Kutta-Fehlberg method. These ideas are implemented in a Fortran program called GERK, which is ACM Algomthm 504.

The art of writing a Runge-Kutta code. II.

All those factors bearing on the creation of a high quality Runge-Kutta code for the solution of ordinary differential equations are studied. A code, RKF45, is presented as a concrete realization of the conclusions of the study.

The Art of Writing a Runge-Kutta Code, Part I

Publisher Summary This chapter discusses Runge–Kutta methods of moderate order and their implementation as a computer code. There is a widely held belief that writing a Runge-Kutta code is not much more than a transcription of the basic formula into a high level computer language. One should restrict the attention to methods that are of order four but because of local extrapolation, this may also involve the formulas of orders three and five. A four-stage fourth order formula of Kutta is known as the classical scheme and has been extremely popular because of its simple coefficients. This principle involves taking two half steps with the basic formula and combining the result with that from one whole step of the basic formula so as to estimate the error of the more accurate result. The most important single aspect of a formula is its accuracy for it is the main factor affecting efficiency. One should not try to solve truly stiff problems with explicit Runge–Kutta formulas; however, some stiffness is not unusual. The essence of the matter is that the step size is limited for the reasons of stability in the presence of stiffness rather than for the reasons of accuracy that is more commonly the case.

Comparing Error Estimators for Runge-Kutta Methods

A way is proposed to compare local error estimators. This is applied to the major estimators for fourth-order Runge-Kutta procedures. An estimator which leads to a production code 18% more efficient than a code using the standard one is recommended and supported by numerical tests. 1. Introduction. The design of a production code for the numerical solution of ordinary differential equations requires three major choices (at least): the method of advancing the solution one step, the method of estimating the local error incurred in one step, and the strategy for choosing the next step length. The first choice has been intensively studied in the literature of numerical analysis but the latter two have been rather neglected in spite of their great practical significance. The aim of this paper is to compare alternatives for estimating the local error. We restrict ourselves to estima- tors appropriate to fourth-order Runge-Kutta methods for two reasons. Production codes based on fourth-order Runge-Kutta processes are quite common and we are able to recommend improvements based on this study. Secondly, texts frequently mention how easy it is to change the step size when using Runge-Kutta procedures but they often fail to point out the associated practical disadvantages. Namely, it is rather difficult and expensive to estimate the local error and so decide when to change the step size. Furthermore, the standard estimator forces one to use a relatively crude step size strategy. It is, therefore, important to consider the reliability and efficiency of error estimating schemes. In Section 2, we shall propose a way of comparing numerically local error estima- tors which seems to be of broad applicability. Section 3 describes the most useful error estimators known to the authors, which are then compared in the fourth section.

Algorithm 504: GERK: Global Error Estimation For Ordinary Differential Equations [D]

SUBROUTINE GERK(F, NEQN, Y, T, TOUT, RELERR, ABSERR, IFLAG, GER 10 * GERROR, WORK, IWORK) GER 20 GER 30 FEHLBERG FOURTH(FIFTH) ORDER RUNGE-KUTTA METHOD WITH GER 40 GLOBAL ERROR ASSESSMENT GER 50 GER 60 WRITTEN BY H.A.WATTS AND L.F.SHAMPINE GER 70 SANDIA LABORATORIES GER 88 GER 90 GERK IS DESIGNED TO SOLVE SYSTEMS OF DIFFERENTIAL EQUATIONS GER 1@O WHEN IT IS IMPORTANT TO HAVE A READILY AVAILABLE GLOBAL ERROR GER 110 ESTIMATE. PARALLEL INTEGRATION IS PERFORMED TO YIELD TWO GER 120 SOLU7IONS ON DIFFERENT MESH SPACINGS AND GLOBAL EXTRAPOLATION GER 130 IS APPLIED TO PROVIDE AN ESTIMATE OF THE GLOBAL ERROR IN THE GER 140

Solving complex-valued differential systems

Linear two-point boundary-value problems defined in complex space arise frequently in physical studies. We examine initial-value procedures based on superposition principles applied to the associated real system of equations. We show how to take advantage of the fact that the underlying problem is complex, thereby cutting the usual number of necessary integrations of the homogeneous system by one-half. We also discuss several aspects of incorporating these ideas into a computer code utilizing an orthonormalization procedure.

COMPARISON OF SOME CODES FOR THE INITIAL VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS

Publisher Summary This chapter presents a comparison of some codes for the initial value problem for ordinary differential equations. Several techniques for solving boundary value problems such as shooting and invariant imbedding require the use of codes, which solve the initial value problem. Hence, it is important to have available good quality software for this purpose. The chapter presents some preliminary results of a study comparing the performance of several first class general purpose codes for solving the initial value problem for non-stiff ordinary differential equations. For low accuracy requests, RKF is probably best as it is competitive with DE/STEP, INTRP, and DVDQ on function counts while also having the smallest overhead cost. For medium accuracy requests, the choice between RKF and DE/STEP, INTRP, or DVDQ depends on how expensive the derivative evaluations are, and for high accuracy, requests DE/STEP, INTRP, or DVDQ is usually be the best choice. DIFSUB is generally not competitive in any range of tolerances. EXTRAP is considered competitive at high-accuracy requests when the derivative evaluations are inexpensive.

A smooth output interpolation process for BDF codes

The original line of codes developed and popularized by C.W. Gear and A.C. Hindmarsh (which use the backward differentiation formulae appropriate for solving stiff problems) produce discontinuous derivative approximations at all mesh points and discontinuous solution approximations at certain mesh points when a change in the order of the method takes place. We discuss the changes necessary to eliminate these discrepancies and define a globally smooth interpolation process. These improvements are particularly important in the context of defining a mathematically sound root solving algorithm.

COMPUTATION OF EIGENVALUES/EIGENFUNCTIONS FOR TWO POINT BOUNDARY VALUE PROBLEMS

Publisher Summary This chapter describes the computation of eigenvalues/eigenfunctions for two point boundary value problem. In solving linear two-point boundary value problems, the code SUPORT uses the method of superposition together with orthonormalization of the base solutions to the homogeneous equation when linear dependence threatens. This chapter presents extensions of the procedure used by SUPORT to allow computation of solutions of eigenvalue problems. The technique is iterative on the eigenvalue parameter and requires a nonlinear equation solver as a driver routine. When solving eigenvalue problems by means of an initial value technique, the nonlinear function evaluated by the root finder is dependent on the boundary conditions at the final end point. The chapter describes the effects of the orthonormalization process on the iterative scheme and the advantages of preassigning orthonormalization points. It presents several examples that demonstrate the applicability of the code and allow comparison of various optional features.