Wayne H. Enright

Comparing numerical methods for stiff systems of O.D.E:s

This paper describes a technique for comparing numerical methods that have been designed to solve stiff systems of ordinary differential equations. The basis of a fair comparison is discussed in detail. Measurements of cost and reliability are made over a collection of 25 carefully selected problems. The problems have been designed to show how certain major factors affect the performance of a method.The technique is applied to five methods, of which three turn out to be quite good, including one based on backward differentiation formulas, another on second derivative formulas, and a third on extrapolation. However, each of the three has a weakness of its own, which can be identified with particular problem characteristics.

Interpolants for Runge-Kutta formulas

A general procedure for the construction of interpolants for Runge-Kutta (RK) formulas is presented. As illustrations, this approach is used to develop interpolants for three explicit RK formulas, including those employed in the well-known subroutines RKF45 and DVERK. A typical result is that no extra function evaluations are required to obtain an interpolant with <italic>O</italic>(<italic>h</italic><supscrpt>5</supscrpt>) local truncation error for the fifth-order RK formula used in RKF45; two extra function evaluations per step are required to obtain an interpolant with <italic>O</italic>(<italic>h</italic><supscrpt>6</supscrpt>) local truncation error for this RK formula.

Two FORTRAN packages for assessing initial value methods

We present a discussion and description of a collection of FORTRAN routines designed to aid in the assessment of initial value methods for ordinary differential equations. Although the overall design characteristics are similar to those of earlier testing packages [2,6] that were used for the comparison of methods [5,7], the details and objectives of the current collection are quite different. Our principal objective is the development of testing tools that can be used to assess the efficiency and reliability of a standard numerical method without requiring significant modifications to the method and without the tools themselves affecting the performance of the method.

Robust and reliable defect control for Runge-Kutta methods

The quest for reliable integration of initial value problems (IVPs) for ordinary differential equations (ODEs) is a long-standing problem in numerical analysis. At one end of the reliability spectrum are fixed stepsize methods implemented using standard floating point, where the onus lies entirely with the user to ensure the stepsize chosen is adequate for the desired accuracy. At the other end of the reliability spectrum are rigorous interval-based methods, that can provide provably correct bounds on the error of a numerical solution. This rigour comes at a price, however: interval methods are generally two to three orders of magnitude more expensive than fixed stepsize floating-point methods. Along the spectrum between these two extremes lie various methods of different expense that estimate and control some measure of the local errors and adjust the stepsize accordingly. In this article, we continue previous investigations into a class of interpolants for use in Runge-Kutta methods that have a defect function whose qualitative behavior is asymptotically independent of the problem being integrated. In particular the point, in a step, where the maximum defect occurs as h → 0 is known a priori. This property allows the defect to be monitored and controlled in an efficient and robust manner even for modestly large stepsizes. Our interpolants also have a defect with the highest possible order given the constraints imposed by the order of the underlying discrete formula. We demonstrate the approach on three Runge-Kutta methods of orders 5, 6, and 8, and provide Fortran and preliminary Matlab interfaces to these three new integrators. We also consider how sensitive such methods are to roundoff errors. Numerical results for four problems on a range of accuracy requests are presented.

Improving the Efficiency of Matrix Operations in the Numerical Solution of Stiff Ordinary Differential Equations

In the numerical solution of large stiff systems of ordinary differential equations, matrix operations associated with the solution of linear equations often dominate the solution time. A matrix factorization is suggested that will allow efficient updating after a change in stepsize or order. This updating technique is shown to be applicable to a wide variety of methods for stiff systems including multistep methods, Runge-Kutta methods, and methods using a rational function of a matrix The technique is particularly useful if the system is large and the Jacobian is dense Numerical results are included to illustrate the use of the technique.

Runge-Kutta Software with Defect Control four Boundary Value ODEs

A popular approach to the numerical solution of boundary value ODE problems involves the use of collocation methods. Such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval. On the other hand, several authors have suggested as an alternative, certain subclasses of the implicit Runge--Kutta formulas, known as mono-implicit Runge--Kutta (MIRK) formulas, which can be implemented at a lower cost per step than the collocation methods. These latter formulas do not have a natural implementation that provides a continuous approximation to the solution; rather, only a discrete approximation at certain points within the problem interval is obtained. However, recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit Runge--Kutta formula. These ideas have recently been extended to develop continuous extensions of the MIRK formulas. In this paper, we describe our investigation of the use of continuous MIRK formulas in the numerical solution of boundary value ODE problems. A primary thrust of this investigation is to consider defect control, based on the continuous MIRK formulas, as an alternative to the standard use of global error control, as the basis for termination and mesh redistribution criteria.

A delay differential equation solver based on a continuous Runge–Kutta method with defect control

We have recently developed a generic approach for solving neutral delay differential equations based on the use of a continuous Runge–Kutta formula with defect control and investigated its convergence properties. In this paper, we describe a method, DDVERK, which implements this approach and justify the strategies and heuristics that have been adopted. In particular we show how the assumptions related to error control, stepsize control, and discontinuity detection (required for convergence) can be efficiently realized for a particular sixth-order numerical method. Summaries of extensive testing are also reported.

Continuous numerical methods for ODEs with defect control

Abstract Over the last decade several general-purpose numerical methods for ordinary differential equations (ODEs) have been developed which generate a continuous piecewise polynomial approximation that is defined for all values of the independent variable in the range of interest. For such methods it is possible to introduce measures of the quality of the approximate solution based on how well the piecewise polynomial satisfies the ODE. This leads naturally to the notion of “defect-control”. Numerical methods that adopt error estimation and stepsize selection strategies in order to control the magnitude of the associated defect can be very effective and such methods are now being widely used. In this paper we will review the advantages of this class of numerical methods and present examples of how they can be effectively applied. We will focus on numerical methods for initial value problems (IVPs) and boundary value problems (BVPs) where most of the developments have been introduced but we will also discuss the implications and related developments for other classes of ODEs such as delay differential equations (DDEs) and differential algebraic equations (DAEs).

Continuous Runge-Kutta methods for neutral Volterra integro-differential equations with delay

Abstract We investigate explicit and implicit continuous Runge-Kutta methods for solving neutral Volterra integro-differential equations with delay. We consider the convergence of the iterative scheme required on each step and the convergence of the numerical solution to the true solution of the Volterra systems.

Effective solution of discontinuous IVPs using a Runge-Kutta formula pair with interpolants

An automatic technique for solving discontinuous initial-value problems is developed and justified. The technique is based on the use of local interpolants such as those that have been developed for use with Runge-Kutta formula pairs. Numerical examples are presented to illustrate the significant improvement in efficiency and reliability that results when this technique is used with standard methods.