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.

Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here we give a brief overview of differential equations that can now be solved by R.

Recently there has been considerable interest in the approximate numerical integration of the special initial value problem y'=f(x,y) for cases where it is known in advance that the required solution is periodic. The well known class of St6rmer-Cowett methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that of P-stability. However Lambert and Watson have shown that a P-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems. P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.

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Summary.A class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived. An algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiation scheme of higher order. This approach allows us to develop L-stable schemes of order up to 4 and L(cQ-stable schemes of order up to 9. An algorithm based on the integration formulae derived in this paper is illustrated by some numerical examples and it is shown that it is often superior to certain existing algorithms.