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

AbstractFor 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

Multistep variable methods for exact integration of perturbed stiff linear systems

A family of real and analytical functions with values within the ring of M(m, R) is introduced. The solution for linear systems of differential equations is expressed as a series of Φ-functions. This new multistep method is defined for variable-step and variable-order, maintains the good properties of the Φ-function series method. It incorporates to compute the coefficients of the algorithm a recurrent algebraic procedure, based in the existing relation between the divided differences and the elemental and complete symmetrical functions. In addition, under certain hypotheses, the new multistep method calculates the exact solution of the perturbed problem. The new method is implemented in a computational algorithm which enables us to resolve in a general manner some physics and engineering IVP’s modeled by means systems of differential equations. The good behaviour and precision of the method is evidenced by contrasting the results with other-reputed algorithms and even with methods based on Scheifele’s G-functions.This work has been supported by GRE09-13 project of the University of Alicante and the project of the Generalitat Valenciana GV/2011/032

Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems

On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed.Comment: 25 pages, 4 figures (typos fixed

A class of explicit two-step hybrid methods for second-order IVPs

AbstractA class of explicit two-step hybrid methods for the numerical solution of second-order IVPs is presented. These methods require a reduced number of stages per step in comparison with other hybrid methods proposed in the scientific literature. New explicit hybrid methods which reach up to order five and six with only three and four stages per step, respectively, and which have optimized the error constants, are constructed. The numerical experiments carried out show the efficiency of our explicit hybrid methods when they are compared with classical Runge–Kutta–Nyström methods and other explicit hybrid codes proposed in the scientific literature

Two point block multistep methods with trigonometric−fitting for solving oscillatory problems

In this paper, we present the absolute stability of the existing 2-point implicit block multistep step methods of step number k = 3 and k = 5 and solving special second order ordinary differential equations (ODEs). The methods are then trigonometrically fitted so that they are suitable for solving highly oscillatory problems arising from the special second order ODEs. Their explicit counterparts are also trigonometrically fitted so that in the implementation the methods can act as a predictor-corrector pairs. The numerical results based on the integration over a large interval are given to show the performance of the proposed methods. From the numerical results we can conclude that the new trigonometrically-fitted methods are superior in terms of accuracy and execution time, compared to the existing methods in the scientific literature when used for solving problems which are oscillatory in nature

SARK: a type-insensitive Runge-Kutta code

A novel solution method based on Mono-implicit Runge-Kutta methods has been fully developed and analysed for the numerical solution of stiff systems of ordinary differential equations (ODE). These Backward Runge-Kutta (BRK) methods have very desirable stability properties which make them efficient for solving a certain class of ODE which are not solved adequately by current methods. These stability properties arise from applying a numerical method to the standard test problem and analysing the resulting stability function. This technique, however, fails to show the full potential of a method. With this in mind a new graphical technique has been derived that examines the methods performance on the standard test case in much greater detail. This technique allows a detailed investigation of the characteristics required for a numerical integration of highly oscillatory problems. Numerical ODE solvers are used extensively in engineering applications, where both stiff and non-stiff systems are encountered, hence a single code capable of integrating the two categories, undetected by the user, would be invaluable. The BRK methods, combined with explicit Runge-Kutta (ERK) methods, are incorporated into such a code. The code automatically determines which integrator can currently solve the problem most efficiently. A switch to the most efficient method is then made. Both methods are closely linked to ensure that overheads expended in the switching are minimal. Switching from ERK to BRK is performed by an existing stiffness detection scheme whereas switching from BRK to ERK requires a new numerical method to be devised. The new methods, called extended BRK (EBRK) methods, are based on the BRK methods but are chosen so as to possess stability properties akin to the ERK methods. To make the code more flexible the switching of order is also incorporated. Numerical results from the type-insensitive code, SARK, indicate that it performs better than the most widely used non-stiff solver and is often more efficient than a specialized stiff solver

Diagonally implicit hybrid method for solving special second order ordinary differential equations

This paper describes the derivation of a fifth-order diagonally implicit hybrid method. The method is zero dissipative and has phase-lag of order six. The method is compared with the existing hybrid method and the numerical comparisons carried out show that the new method improves the accuracy of the existing method for solving several special second order ordinary differential equations