Jeff R. Cash

A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides

Explicit Runge-Kutta methods (RKMs) are among the most popular classes of formulas for the approximate numerical integration of nonstiff, initial value problems. However, high-order Runge-Kutta methods require more function evaluations per integration step than, for example, Adams methods used in PECE mode, and so, with RKMs, it is expecially important to avoid rejected steps. Steps are often rejected when certain derivatives of the solutions are very large for part of the region of integration. This corresponds, for example, to regions where the solution has a sharp front or, in the limit, some derivative of the solution is discontinuous. In these circumstances the assumption that the local truncation error is changing slowly is invalid, and so any step-choosing algorithm is likely to produce an unacceptable step. In this paper we derive a family of explicit Runge-Kutta formulas. Each formula is very efficient for problems with smooth solution as well as problems having rapidly varying solutions. Each member of this family consists of a fifty-order formula that contains imbedded formulas of all orders 1 through 4. By computing solutions at several different orders, it is possible to detect sharp fronts or discontinuities before all the function evaluations defining the full Runge-Kutta step have been computed. We can then either accpet a lower order solution or abort the step, depending on which course of action seems appropriate. The efficiency of the new algorithm is demonstrated on the DETEST test set as well as on some difficult test problems with sharp fronts or discontinuities.

A deferred correction method for nonlinear two-point boundary value problems: implementation and numerical evaluation

A deferred correction method for the numerical solution of nonlinear two-point boundary value problems has been derived and analyzed in two recent papers by the first author. The method is based on mono-implicit Runge–Kutta formulas and is specially designed to deal efficiently with problems whose solutions contain nonsmooth parts—in particular, singular perturbation problems of boundary layer or turning point type. This paper briefly describes an implementation of the method and gives the results of extensive numerical testing on a set of nonlinear problems that includes both smooth and increasingly stiff (and difficult) problems. Results on the test set are also given using the available codes COLSYS and COLNEW. Although the intent is not to make a formal comparison, the code described appears to be competitive in speed and storage requirements on these problems.

A Deferred Correction Method for Nonlinear Two-Point Boundary Value Problems

A deferred correction method for the numerical solution of nonlinear two-point boundary value problems has been derived and analyzed in two recent papers by the first author. The method is based on mono-implicit Runge–Kutta formulas and is specially designed to deal efficiently with problems whose solutions contain nonsmooth parts—in particular, singular perturbation problems of boundary layer or turning point type. This paper briefly describes an implementation of the method and gives the results of extensive numerical testing on a set of nonlinear problems that includes both smooth and increasingly stiff (and difficult) problems. Results on the test set are also given using the available codes COLSYS and COLNEW. Although the intent is not to make a formal comparison, the code described appears to be competitive in speed and storage requirements on these problems.

An MEBDF code for stiff initial value problems

In two recent papers one of the present authors has proposed a class of modified extended backward differentiation formulae for the numerical integration of stiff initial value problems. In this paper we describe a code based on this class of formulae and discuss its performance on a large set of stiff test problems.

Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs

Abstract For many years the methods of choice for the numerical solution of stiff initial value problems and certain classes of differential algebraic equations have been the well-known backward differentiation formulae (BDF). More recently, however, new classes of formulae which can offer some important advantages over BDF have emerged. In particular, some recent large-scale independent comparisons have indicated that modified extended backward differentiation formulae (MEBDF) are particularly efficient for general stiff initial value problems and for linearly implicit DAEs with index ⩽3. In the present paper we survey some of the more important theory associated with these formulae, discuss some of the practical applications where they are particularly effective, e.g., in the solution of damped highly oscillatory problems, and describe some significant recent extensions to the applicability of MEBDF codes.

Towards efficient Runge-Kutta methods for stiff systems

A special class of implicit Runge–Kutta methods is developed for the numerical solution of stiff initial value problems. These formulae are derived from known singly implicit methods by adding one or more extra diagonally implicit stages. It is hoped that this modification of the original method will lead to an overall gain in efficiency, and an analysis of the advantages of making this enhancement is presented.

A Class of Implicit Runge-Kutta Methods for the Numerical Integration of Stiff Ordinary Differential Equations

One-step methods similar in design to the well-known class of Runge-Kutta methods are developed for the efficient numerical integration of both stiff and nonstiff systems of first-order ordinary differential equations The algomthms developed combine accuracy in the hrait h --~ 0 with a large regmn of absolute stabdity and are demonstrated by direct apphcation to certain particular examples.

An automatic continuation strategy for the solution of singularly perturbed nonlinear boundary value problems

In a recent paper, the present authors derived an automatic continuation algorithm for the solution of linear singular perturbation problems. The algorithm was incorporated into two general-purpose codes for solving boundary value problems, and it was shown to deal effectively with a large test set of linear problems. The present paper describes how the conintuation algorithm for linear problems can be extended to deal with the nonlinear case. The results of exstensive numerical testing on a set of nonlinear singular perturbation problems are given, and these clearly demonstrate the efficacy of continuation for solving such problems.

Implementation issues in solving nonlinear equations for two-point boundary value problems

Complex numerical methods often contain subproblems that are easy to state in mathematical form, but difficult to translate into software. Several algorithmic isues of this nature arise in implementing a Newton iteration scheme as part of a finite-difference method for two-point boundary value problems. We describe the practical as well as theoretical considerations behind the decisions included in the final code, with special emphasis on two “watchdog” strategies designed to improve reliability and allow early termination of the Newton iterates.ZusammenfassungKomplizierte numerische Methoden enthalten oft Teilprobleme, die sich leicht mathematisch formulieren lassen, die aber schwierig in Computerprogramme umgewandelt werden können. Mehrere solche Fragestellungen traten bei der Erstellung eines Newton-Verfahrens als Bestandteil eines Differenzenverfahrens für 2-Punkt Randwert-probleme auf. Wir beschreiben die praktischen und theoretischen Überlegungen, welche den Entscheidungen zugrunde liegen, die schließlich zum Computerprogramm führten. Insbesondere betonen wir dabei zwei “Watchdog”-Strategien, welche die Zuverlässigkeit verbessern und ein frühes Abbrechen der Newton-Iteration ermöglichen.

Mesh selection for stiff two-point boundary value problems

This paper is concerned with the mesh selection algorithm of COLSYS, a well known collocation code for solving systems of boundary value problems. COLSYS was originally designed to solve non-stiff and mildly stiff problems only. In this paper we show that its performance for solving extremely stiff problems can be considerably improved by modifying its error estimation and mesh selection algorithms. Numerical examples indicate the superiority of the modified algorithm.