Higinio Ramos

An optimized two-step hybrid block method for solving general second order initial-value problems

A new optimized two-step hybrid block method for the numerical integration of general second-order initial value problems is presented. The method considers two intra-step points which are selected adequately in order to optimize the local truncation errors of the main formulas for the solution and the derivative at the final point of the block. The new method is zero-stable and consistent with fifth algebraic order. Numerical experiments used revealed the superiority of the new method for solving this kind of problems, in comparison with methods of similar characteristics in the literature.

On the frequency choice in trigonometrically fitted methods

The choice of frequency in trigonometrically fitted methods is a fundamental question, especially if long-term prediction is considered. For linear oscillators, the frequency of the method is the same as the frequency of the solution of the differential equation. However, for nonlinear problems the frequency of the method is, in general, different from the frequency of the true solution. We present some experiments showing how the frequency depends strongly on certain values.

A non-standard explicit integration scheme for initial-value problems

In this paper we present the construction of a non-standard explicit algorithm for initial-value problems. The method results to be of second order and A-stable. This new algorithm has been proven to be suitable for solving different kind of initial-value problems, specifically, non-singular problems, singular problems, stiff problems and singularly perturbed problems. Some numerical experiments are considered in order to check the behaviour of the method when applied to a variety of initial-value problems.

Exponential fitting BDF-Runge-Kutta algorithms

In other papers, the authors presented exponential fitting methods of BDF type. Now, these methods are used to derive some BDF–Runge–Kutta type formulas (of second-, third- and fourth-order), capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential with parameter A and ordinary polynomials. Theorems of the truncation error reveal the good behavior of the new methods for stiff problems. Plots of their absolute stability regions that include the whole of the negative real axis are provided. Different procedures to find the parameter of the method are proposed, using these techniques there will not be necessary to compute the exponential matrix at each step, even when nonlinear problems are integrated. Numerical examples underscore the efficiency of the proposed codes, especially when they are integrating stiff problems.

Variable stepsize störmer-cowell methods

Stormer and Cowell methods are multistep codes for the numerical solution of second-order initial value problems where the first derivative does not appear explicitly. In this paper, we develop a procedure to obtain k-step Stormer and Cowell methods in their variable step size version, avoiding computation of the coefficients by recurrences or integrals. We offer a strategy for conveniently selecting the stepsize. Considering a pair of explicit and implicit formulae, these may be implemented in a predictor-corrector mode.

On the choice of the frequency in trigonometrically-fitted methods for periodic problems

In this paper the use of a trigonometrically fitted method to obtain the approximate solutions of some nonlinear periodic oscillators is presented. A great number of different approaches have been considered to obtain analytical approximations for this kind of problems: a generalized decomposition method (GDM), a linearized harmonic balance procedure (LHB), the homotopy perturbation method (HPM), the harmonic balanced method (HBM), the Adomian decomposition method, etc. From those approaches, analytical approximations to the frequency of oscillation and periodic solutions are obtained, which are valid for a large range of amplitudes of oscillation. However, these techniques have been limited to obtain only one or two iterates because of the great amount of algebra involved. We use a trigonometrically adapted method to obtain numerical approximations to the solutions, yielding very acceptable results, on the basis that the approximation of the frequency of the method is done with great accuracy. There are a lot of trigonometrically fitted methods in the literature, but there is not a definite way to obtain the optimal value of the frequency. We present a strategy for the choice of the parameter value in the adapted method based on the minimization of the total energy. Some examples solved numerically confirm the good performance of the adopted strategy.

Review of explicit Falkner methods and its modifications for solving special second-order I.V.P.s

The unusual implementation of explicit Falkner methods for solving special second-order initial-value problems increases the order of the method as will be shown by means of numerical examples. In this paper we made an analysis of the propagation of the truncation errors in both the usual and the unusual implementations, thus justifying the numerical results obtained. By that, some error bounds for the global truncation errors on the solution and on the derivative are provided. Some numerical examples confirm that the bounds are realistic. Stability analysis is also addressed, and intervals of stability are presented.

A trigonometrically-fitted method with two frequencies, one for the solution and another one for the derivative

Abstract In trigonometrically-fitted methods the determination of the parameter (usually known as the frequency) is a critical issue, as was shown in the article by H. Ramos and J. Vigo-Aguiar [Applied Mathematics Letters, 23 (2010) 1378–1381]. If the frequency estimation relies on the vanishing of the principal term of the local truncation error, then the first derivative is present in the formula for approximating the parameter. This requires the use of a procedure for approximating the first derivative. For this purpose we use another trigonometrically-fitted formula with a second parameter (different from the frequency of the principal method and also different from the frequency of the true solution). We describe how to approximate both parameters on each step and present different experiments concerning these questions. The numerical results indicate the good performance of the strategy.

A new algorithm appropriate for solving singular and singularly perturbed autonomous initial-value problems

A nonlinear explicit scheme is proposed for numerically solving first-order singular or singularly perturbed autonomous initial-value problems (IVP) of the form y ′=f(y). The algorithm is based on the local approximation of the function f(y) by a second-order Taylor expansion. The resulting approximated differential equation is then solved without local truncation error. For the true solution the method has a local truncation error that behaves like either 𝒪(h 3) or 𝒪(h 4) according to whether or not some parameter vanishes. Some numerical examples are provided to illustrate the performance of the method. Finally, an application of the method for detecting and locating singularities is outlined.

A numerical ODE solver that preserves the fixed points and their stability

In this paper, we provide a one-step predictor-corrector method for numerically solving first-order differential initial-value problems with two fixed points. The method preserves the stability behaviour of the fixed points, which results in an efficient integrator for this kind of problem. Some numerical examples are provided to show the good performance of the method.