On the stability properties of Brown's multistep multiderivative methods

Summary Brown introducedk-step methods usingl derivatives. We investigate for whichk andl the methods are stable or unstable. It is seen that to anyl the method becomes unstable fork large enough. All methods withk2(l+1) are stable. Fork=1,2,..., 18 there exists a _k such that the methods are stable for anyl _k and unstable for anyl < _k . The _k are given.     

Stability properties of variable stepsize variable formula methods

Summary When variable stepsize variable formula methods (VSVFM's) are used in the solution of systems of first order differential equations instability arises sometimes. Therefore it is important to find VSVFM's whose zerostability properties are not affected by the choice of both the stepsize and the formula. The Adams VSVFM's are such methods. In this work a more general class of methods which contains the Adams VSVFM's is discussed and it is proved that the zero-stability of the class is not affected by the choice of the stepsize and of the formula.     

On the order of composite multistep methods for ordinary differential equations

Summary In this paper, a general class ofk-step methods for the numerical solution of ordinary differential equations is discussed. It is shown that methods with order of consistencyq have order of convergence (q+1) if a very simple condition is satisfied. This result gives a new aspect to previous results of Spijker; it also serves as a starting point for a new theory of cyclick-step methods, completing an approach of Donelson and Hansen. It facilitates the practical determination of high-order cyclick-step methods, especially of stiffly stable,k-step methods.     

High order methods for the numerical integration of ordinary differential equations

Summary High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.     

Partitioned adaptive Runge-Kutta methods and their stability

Summary This paper deals with the solution of partitioned systems of nonlinear stiff differential equations. Given a differential system, the user may specify some equations to be stiff and others to be nonstiff. For the numerical solution of such a system partitioned adaptive Runge-Kutta methods are studied. Nonstiff equations are integrated by an explicit Runge-Kutta method while an adaptive Runge-Kutta method is used for the stiff part of the system.The paper discusses numerical stability and contractivity as well as the implementation and usage of such compound methods. Test results for three partitioned stiff initial value problems for different tolerances are presented.     

A study of Rosenbrock-type methods of high order

Summary This paper deals with the solution of nonlinear stiff ordinary differential equations. The methods derived here are of Rosenbrock-type. This has the advantage that they areA-stable (or stiffly stable) and nevertheless do not require the solution of nonlinear systems of equations. We derive methods of orders 5 and 6 which require one evaluation of the Jacobian and oneLU decomposition per step. We have written programs for these methods which use Richardson extrapolation for the step size control and give numerical results.     

Adams methods for neutral functional differential equations

Summary In this paper Adams type methods for the special case of neutral functional differential equations are examined. It is shown thatk-step methods maintain orderk+1 for sufficiently small step size in a sufficiently smooth situation. However, when these methods are applied to an equation with a non-smooth solution the order of convergence is only one. Some computational considerations are given and numerical experiments are presented.     

Local error estimation for multistep collocation methods

Multistep collocation methods for initial value problems in ordinary differential equations are known to be a subclass of multistep Runge-Kutta methods and a generalisation of the well-known class of one-step collocation methods as well as of the one-leg methods of Dahlquist. In this paper we derive an error estimation method of embedded type for multistep collocation methods based on perturbed multistep collocation methods. This parallels and generalizes the results for one-step collocation methods by Nørsett and Wanner. Simple numerical experiments show that this error estimator agrees well with a theoretical error estimate which is a generalisation of an error estimate first derived by Dahlquist for one-leg methods.     

One-step methods of any order for ordinary differential equations with discontinuous right-hand sides

Summary The right-hand sides of a system of ordinary differential equations may be discontinuous on a certain surface. If a trajectory crossing this surface shall be computed by a one-step method, then a particular numerical analysis is necessary in a neighbourhood of the point of intersection. Such an analysis is presented in this paper. It shows that one can obtain any desired order of convergence if the method has an adequate order of consistency. Moreover, an asymptotic error theory is developed to justify Richardson extrapolation. A general one-step method is constructed satisfying the conditions of the preceding theory. Finally, a simplified Newton iteration scheme is used to implement this method.     

Order conditions for numerical methods for partitioned ordinary differential equations

Summary Motivated by the consideration of Runge-Kutta formulas for partitioned systems, the theory of P-series is studied. This theory yields the general structure of the order conditions for numerical methods for partitioned systems, and in addition for Nyström methods fory=f(y,y), for Rosenbrock-type methods with inexact Jacobian (W-methods). It is a direct generalization of the theory of Butcher series [7, 8]. In a later publication, the theory ofP-series will be used for the derivation of order conditions for Runge-Kutta-type methods for Volterra integral equations [1].     

Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

Summary GeneralizedA()-stable Runge-Kutta methods of order four with stepsize control are studied. The equations of condition for this class of semiimplicit methods are solved taking the truncation error into consideration. For application anA-stable and anA(89.3°)-stable method with small truncation error are proposed and test results for 25 stiff initial value problems for different tolerances are discussed.     

A note onB-stability of Runge-Kutta methods

Summary Burrage and Butcher [1, 2] and Crouzeix [4] introduced for Runge-Kutta methods the concepts ofB-stability,BN-stability and algebraic stability. In this paper we prove that for any irreducible Runge-Kutta method these three stability concepts are equivalent.Chapters 1–3 of this article have been written by the second author, whereas chapter 4 has been written by the first author     

A 0-Stability and stiff stability of Brown's multistep multiderivative methods

Summary Brown [1] introducedk-step methods usingl derivatives. Necessary and sufficient conditions forA ₀-stability and stiff stability of these methods are given. These conditions are used to investigate for whichk andl the methods areA ₀-stable. It is seen that for allk andl withk1.5 (l+1) the methods areA ₀-stable and stiffly stable. This result is conservative and can be improved forl sufficiently large. For smallk andl A ₀-stability has been determined numerically by implementing the necessary and sufficient condition.     

On the stability of backward differentiation methods

Summary The approximation of linear systemsy=–A(t)y+b(t) by backward differentiation methods up to order 5 is considered. It is proved that the error does not increase if the real symmetric matrixA(t) is positive definite andA(t) is negative semi-definite.     

Liapunov functions and error bounds for approximate solutions of ordinary differential equations

Summary It is shown that Liapunov functions may be used to obtain error bounds for approximate solutions of systems of ordinary differential equations. These error bounds may reflect the behaviour of the error more accurately than other bounds.     

High orderP-stable formulae for the numerical integration of periodic initial value problems

Summary Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Störmer-Cowell 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 ofP-stability. However Lambert and Watson have shown that aP-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.     

On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae

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 developL-stable schemes of order up to 4 andL()-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.     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.     

Unconditionally stable explicit methods for parabolic equations

Summary This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA ₀-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.     

Convergence of multistep methods for Volterra functional differential equations

Summary This paper deals with the convergence of nonstationary quasilinear multistep methods with varying step, used for the numerical integration of Volterra functional differential equations. A Perron type condition (appearing in the differential equations theory) is imposed on the increment function. This gives a generalization of some results of Tavernini ([19–21]).